Quantum mechanics is often developed in the position representation, but this is not necessary, and one can perform calculations in a representation-independent fashion, even for wavefunctions. In this work, we illustrate how one can determine wavefunctions, aside from normalization, using only operators and how those operators act on state vectors. To do this in plane polar and spherical coordinates requires one to convert the translation operator into those coordinates. As examples of this approach, we illustrate the solution of the Coulomb problem in two and three dimensions without needing to express any operators in position space.

The quantum-mechanical position-space translation operator

T̂(r)=eirp̂=eirxp̂x+ryp̂y+rzp̂z
(1)

is normally expressed in terms of Cartesian components,

r=exrx+eyry+ezrz,p̂=exp̂x+eyp̂y+ezp̂z,
(2)

with ex, ey, and ez being the Cartesian unit vectors. We use hats to denote the operators on the state vector space. Note, in particular, that the quantities rα in the translation operator are dimensionful numbers, not operators. The position and momentum operators satisfy the canonical commutation relation

[r̂α,p̂β]=iδα,β,
(3)

where α and β run over the Cartesian dimensions.

The translation operator can be employed to determine position eigenstates by translating the position-space eigenvector at the origin, |0r, which satisfies

r̂α|0r=0.
(4)

One of our assumptions is that such a position eigenvector at the origin exists, without worrying about the details of rigged Hilbert spaces; we will see that the wavefunction at any position can be determined relative to the wavefunction at the origin. We then have the position-operator eigenstates given by

|r=T̂(r)|0r.
(5)

It is easy to verify that

r̂α|r=rα|r
(6)

by using the braiding relation (derived from the Hadamard lemma)

eÂf(B̂)eÂ=fB̂+Â,B̂+12!Â,Â,B̂+13!Â,Â,Â,B̂+,
(7)

which is valid, provided that f(B̂) can be written as a sum of non-negative integral powers of B̂ and the argument of the function on the right-hand side is a sum of terms consisting of increasingly nested commutators. For the verification of the position eigenstate, one simply notes that the commutator of position with momentum is a dimensionful number, and so it commutes with all operators, which truncates the series after the first commutator.

In this work, we discuss how to calculate wavefunctions using a representation-independent approach. This sounds like a contradiction since a wavefunction is the representation of a quantum state vector in a particular basis. However, because each component of the wavefunction is constructed from the inner product between the state ket and a basis bra, one can see that each component is an inner product between two vectors in the state vector space and hence is a representation-independent object. The representation only enters when we compute the components for all basis vectors in the given representation and represent the quantum state as a linear superposition over the basis vectors of the representation. Since the wavefunction (or the set of components of the expansion) is constructed from the representation-independent inner products, it can be found in a representation-independent fashion, as will be illustrated below.

The strategy of the representation-independent approach to quantum mechanics is then to write the position-space wavefunction of a state |ψ⟩ in terms of the position eigenstate at the origin and the translation operator according to

ψ(r)=r|ψ=0r|T̂(r)|ψ.
(8)

This expression provides a route to solely evaluate the wavefunction by manipulating operators. This is because when the position operator acts on the position eigenstate at the origin, it annihilates the state at the origin [see Eq. (4)]. To carry out these calculations for energy eigenfunctions, one needs to factorize the Hamiltonian, in order to determine what happens when the momentum operator acts on the state |ψ⟩, using the methodology of the Schrödinger factorization method.1–3 

We term this approach a representation-independent way to calculate the wavefunctions because one does not need to express the operators used to determine the state vectors in the specific basis of the wavefunctions. Instead, we only employ the commutation relations of the operators to determine the wavefunctions (up to overall normalization). For example, when working with the momentum operator, we only use the canonical commutation relation and how the momentum operator acts on the energy eigenstate. We do not need to use p̂=i, the position-space representation of the momentum operator.

Böhm illustrated an alternate way to calculate wavefunctions in a representation-independent fashion,4 and Merzbacher also used this approach.5 Matrix elements of the position operator between energy eigenstates of the simple harmonic oscillator were employed to determine recurrence relations between energy eigenfunctions of different energy eigenvalues, but at the same position. The recurrence relations were then solved in terms of Hermite polynomials, eventually yielding the usual position-space wavefunctions. The general approach we develop here instead fixes the eigenfunction and relates the value of the wavefunction at the origin to the value of the wavefunction at r. This latter approach can be generalized to many different problems other than the simple harmonic oscillator. It is not clear whether the method developed by Böhm can be extended beyond the simple harmonic oscillator. Moreover, our approach can also be employed for momentum-space wavefunctions.

We illustrate this process below. However, before doing so, we point out the reason why we need to transform the translation operator to spherical and plane polar coordinates. Spherically symmetric problems, such as the Coulomb problem, have wavefunctions that are explicit functions of

r=rx2+ry2+rz2.
(9)

Such functions, for odd powers of r, cannot be expanded in a Maclauren series in the Cartesian position components because expansions about the origin in position space do not exist. Hence, one cannot use the Cartesian-basis representation of the translation operator. A similar argument applies for plane polar coordinates.

In Sec. II, we provide the details of how to convert the translation operator from the expression in terms of Cartesian components of the momentum operator to an expression in terms of the components of the momentum operator in spherical and plane polar coordinates. The exact operator expression can be simplified when it acts on |0r, which is the final form we employ to calculate wavefunctions. In Sec. III, we employ this approach to calculate the position-space wavefunctions for two- and three-dimensional Coulomb problems. This methodology is based on Schrödinger’s factorization method. We conclude in Sec. IV. In the  Appendix, we summarize technical details, primarily related to computing commutators in a representation-independent fashion.

To work with a three-dimensional spherically symmetric system, we express the translation operator, T̂(r), in terms of the spherical coordinates, r, θ, and ϕ; the corresponding position operators, r̂, cosθ̂, sinθ̂, cosϕ̂, sinϕ̂ and p̂r, p̂θ, p̂ϕ, the spherical components of the momentum operator, p̂. We use the spherical components of p̂, instead of the canonical momenta, because this is the cleanest way to decompose the inner product rp̂. Note that p̂r is a canonical momentum operator because it is the quantum analog of the classical momentum conjugate to the radial degree of freedom, whereas the same is not true for p̂θ and p̂ϕ. We can still express the translation operator in terms of p̂r, p̂θ, and p̂ϕ even though p̂r is Hermitian, but not self-adjoint, and thus is not an observable. In addition, the translation operator remains unitary regardless of the coordinate system in terms of which it is expressed.

There is a subtle point associated with the symbol |0r: while r=0 implies that rx = ry = rz = 0, it only implies that r = 0 in spherical coordinates. That is, the values of θ and ϕ are indeterminate until specified by some limiting procedure for how the origin is approached. We will use what we call a “north-pole” state oriented along the positive z axis for definiteness.

It is well known that defining operators corresponding to the angles θ̂ and ϕ̂ is problematic.6 Instead, we define the values corresponding to the position eigenstates expressed in spherical coordinates through the cosine or sine of those operators, which are always well defined in terms of r̂α. In particular,

cosθ̂=rẑr̂,sinθ̂=ρ̂r̂,cosϕ̂=r̂xρ̂,sinϕ̂=r̂yρ̂,
(10)

where

r̂=r̂x2+r̂y2+r̂z2andρ̂=r̂x2+r̂y2.
(11)

These are all well-defined, but care must be taken when these operators act on position eigenstates that approach the origin.

Arguably, the easiest way to go from |0r to |r, using spherical coordinates, is to first translate a distance r in the z-direction, rotate by θ about the y axis, and then rotate by ϕ about the z axis, as illustrated in Fig. 1. Indeed, this is the common way to define the spherical coordinates r, θ, and ϕ. The operator needed to do this is

T̂(r)=eiϕL̂zeiθL̂yeirp̂zeiθL̂yeiϕL̂z,
(12)

and we discuss below the subtleties associated with how it operates on |0r. Here, L̂y and L̂z are the y- and z-components, respectively, of L̂=r̂×p̂, the orbital angular momentum operator.

FIG. 1.

Three-step process to move from the origin to (rx, ry, rz): (top) first translate a distance r along the z axis, (middle) rotate about the y axis by an angle θ, and (bottom) rotate by an angle ϕ about the z axis. One can also reach the same point by rotating directly after the translation in the z-direction to the final point, or one can translate in the θ-direction in the xz plane and then rotate by ϕ, or one can translate in the θ, ϕ direction a distance r directly (not shown). All these alternatives lead to the same final point in space. The operators corresponding to each of these different ways to translate and rotate from the origin to the final point are shown in the main text.

FIG. 1.

Three-step process to move from the origin to (rx, ry, rz): (top) first translate a distance r along the z axis, (middle) rotate about the y axis by an angle θ, and (bottom) rotate by an angle ϕ about the z axis. One can also reach the same point by rotating directly after the translation in the z-direction to the final point, or one can translate in the θ-direction in the xz plane and then rotate by ϕ, or one can translate in the θ, ϕ direction a distance r directly (not shown). All these alternatives lead to the same final point in space. The operators corresponding to each of these different ways to translate and rotate from the origin to the final point are shown in the main text.

Close modal

To establish this result, we use the braiding relation twice. First, we note that

eiθL̂yeirp̂zeiθL̂y=eirsinθp̂x+cosθp̂z
(13)

and then

T̂r=eiϕL̂zeirsinθp̂x+cosθp̂zeiϕL̂z=eirsinθcosϕp̂x+sinθsinϕp̂y+cosθp̂z.
(14)

Since

rx=rsinθcosϕ,ry=rsinθsinϕ,andrz=rcosθ,
(15)

Eq. (14) obviously yields the translation operator as expressed in Cartesian coordinates in Eq. (1).

We use Eqs. (A12) and (A13) to express p̂z in terms of the spherical components of momentum according to

p̂z=p̂ri2r̂cosθ̂p̂θsinθ̂.
(16)

Note that the ordering is important in the second term because sinθ̂ does not commute with p̂θ, but there is no ordering ambiguity with the first term. Equations (12) and (16) allow us to eliminate the Cartesian components of momentum from the translation operator and obtain

T̂(r)=eiϕL̂zeiθL̂yeirp̂ri2r̂cosθ̂p̂θsinθ̂eiθL̂yeiϕL̂z.
(17)

We use Eq. (A13) and

êϕ=exsinϕ̂+eycosϕ̂
(18)

to re-express p̂θ according to

p̂θ=1r̂L̂êϕ+i2cotθ̂
(19)

and thus transform the translation operator into its final form

T̂(r)=eiϕL̂zeiθL̂yeirp̂rir̂cosθ̂L̂êϕr̂sinθ̂eiθL̂yeiϕL̂z.
(20)

We emphasize that this is an operator equality. It is the expression of the translation operator in terms of spherical coordinates.

This operator relation can be rewritten in three other forms, which illustrate the different ways that one can derive the translation operator in spherical coordinates. First, note that L̂z commutes with r̂, cosθ̂, sinθ̂, L̂êϕ, and p̂r (or, more simply, it commutes with p̂z). This means that we can introduce the factor expiϕL̂zexpiϕL̂z just to the left of the expiθL̂y term in Eq. (20) and then move the factor expiϕL̂z to the left through the middle exponential factor, yielding

T̂(r)=eiϕL̂zeiθL̂yeiϕL̂zeirp̂rir̂cosθ̂L̂êϕr̂sinθ̂eiϕL̂zeiθL̂yeiϕL̂z.
(21)

This form of the translation operator expresses it as a similarity transformation of the middle exponential factor with respect to the operator

eiϕL̂zeiθL̂yeiϕL̂z=eiθ(L̂xsinϕ+L̂ycosϕ)
(22)
=eiθeϕL̂,
(23)

which follows from the braiding relation. Note that

eϕL̂=L̂xsinϕ+L̂ycosϕ
(24)

is a linear combination of the angular momentum operators with numbers, not operators, as coefficients because eϕ is not an operator here. The translation operator then becomes

T̂(r)=eiθeϕL̂eirp̂rir̂cosθ̂L̂êϕr̂sinθ̂eiθeϕL̂.
(25)

In this form of the translation operator, we first translate along the z axis a distance r and then rotate by an angle θ about an axis along eϕ, i.e., an axis rotated an angle ϕ counterclockwise from the y axis.

The final two ways we express the operator comes from a simple brute-force substitution. We solve Eqs. (A12)–(A.14) for p̂x, p̂y, and p̂z in terms of p̂r, p̂θ, and p̂ϕ and substitute into Eq. (1). Then, we use the inverse of the braiding relation to remove expiθL̂z to the left and its Hermitian conjugate to the right. We also obtain this form by using the braiding relation to move the expiθL̂y factor (and its conjugate) into the exponent in Eq. (20). This yields

T̂(r)=eiϕL̂zexpirp̂rir̂cos(θ̂θ)1r̂L̂êϕsin(θ̂θ)+p̂rir̂sinθsinθ̂+1r̂L̂êϕsinθcosθ̂(cosϕ̂1)L̂zr̂sinθsinϕ̂sinθ̂eiϕL̂z.
(26)

One can interpret this as a translation along the θ direction in the xz plane, followed by a rotation by ϕ about the z axis. If we use the braiding operation to move the remaining exponential factor into the exponent, we would have a single translation of length r along the θ, ϕ direction. We do not write that other final form explicitly here; it is given by the result in Eq. (26) without the exponential factors on the left and the right and with the substitution ϕ̂ϕ̂ϕ.

As one can immediately see, these forms for the translation operator are equivalent and are easily related by employing the braiding relation in different ways. They also have different physical interpretations for how the translation is performed. Of course, the operators in all four cases are identical and are just expressed in terms of different exponential factors.

The remaining task is to determine the proper limiting procedure to apply this operator on the position eigenstate at the origin. We derive it explicitly for the form of the translation operator given in Eq. (20), but the generalization for any of the other forms is straightforward to work out. Our final result is independent of which form is used.

The strategy is to perform a translation along r so that the indeterminate operators corresponding to cosθ̂ and sinθ̂ [when acting on |0r] can be properly defined. (It turns out that the indeterminacy with respect to ϕ need not be resolved in order to determine the final formula for the translation operator.) To this end, we introduce expiδp̂expiδp̂=1 to the left of T̂(r). The vector δ is an ordinary displacement vector, not an operator, and we choose it to be in the same direction as the vector r in the translation operator. We then move the rightmost exponential factor through T̂(r) to the right, which we can do since it commutes with T̂(r), as is easy to see when T̂(r) is expressed in its Cartesian form in Eq. (1). We then operate on |0r, which yields

|r=eiδp̂T̂(r)eiδp̂|0r.
(27)

This equation can be re-expressed as

|r=eiδp̂T̂(r)|δx,δy,δz.
(28)

Since the two rotation operators on the right-hand side of Eq. (20) first rotate by an angle −ϕ about the z axis and then by an angle −θ about the y axis, we find that the final position eigenstate, after the two rotations, points along the +z axis. In other words, the choice that δ lies in the same direction as r implies that

tanϕ=δyδx
(29)

so that

eiϕL̂z|δx,δy,δz=|δx2+δy2,0,δz
(30)

and further implies that

tanθ=δx2+δy2δz
(31)

so that

eiθL̂yeiϕL̂z|δx,δy,δz=|0,0,δ.
(32)

Here,

δ=δx2+δy2+δz2.
(33)

Note that the ket |0, 0, δ⟩ satisfies

cosθ̂|0,0,δ=r̂zr̂|0,0,δ=|0,0,δ
(34)

and

sinθ̂|0,0,δ=ρ̂r̂|0,0,δ=0.
(35)

Because this is a state oriented along the north pole, the action of cosϕ̂ or sinϕ̂ on this ket is indeterminate, in the sense that one cannot determine the action of cosϕ̂ or sinϕ̂ on this state. Our goal is to simplify the form of the translation operator when it acts on this “north-pole” state. We expand the exponential function expip̂rir̂cosθ̂L̂êϕr̂sinθ̂ in a power series and use the action of cosθ̂ and sinθ̂ on the “north-pole” state [in Eqs. (34) and (35)]. Term by term in the power series expansion, we see that the exponential function simplifies and can be resummed to the form expip̂rir̂ acting on the “north-pole” state. This produces

|r=eiδp̂eiϕL̂zeiθL̂yeirp̂rir̂|0,0,δ.
(36)

At this point, because L̂ commutes with r̂ and p̂r, we can separate the radial and angular degrees of freedom according to

|δx,δy,δz=eiϕL̂zeiθL̂y|0,0,δ=|r=δ|θ,ϕ.
(37)

This result for the label of the state with r = δ arises because we define the radial coordinate eigenstates, |r⟩, to satisfy

r̂|r=r|r,
(38)

with r given by Eq. (9). One can immediately verify that r̂|δx,δy,δz=δ|δx,δy,δz, which establishes the use of the label r = δ in Eq. (37).

We similarly define the angular state, |θ, ϕ⟩, to be the state that satisfies

cosθ̂|θ,ϕ=cosθ|θ,ϕ,sinθ̂|θ,ϕ=sinθ|θ,ϕ,
(39)
cosϕ̂|θ,ϕ=cosϕ|θ,ϕ,sinϕ̂|θ,ϕ=sinϕ|θ,ϕ,
(40)

with one exception. The eigenvalue–eigenvector relations in Eq. (40) cannot be satisfied when the eigenvalues in Eq. (39) correspond to the cases where θ = 0 or θ = π.

Note that the operators r̂x, r̂y, and r̂z cannot operate solely on the state |r⟩, i.e., the domain of those operators lies outside of the space of the eigenstates of the radial-position operator r̂. Similarly, they cannot act on the state |θ, ϕ⟩. Of course, they can act on the tensor-product state |r⟩ ⊗ |θ, ϕ⟩.

However, we can allow the operator r̂ to act only on |r⟩ (and as the identity operator on |θ, ϕ⟩) and cosθ̂ (and the other similar trigonometric operators) act only on |θ, ϕ⟩ (and as the identity on |r⟩), as will be seen next. Consider cosθ̂ acting on the state |r sin θ cos ϕ, r sin θ sin ϕ, r cos θ⟩ = |r⟩ ⊗ |θ, ϕ⟩, which is given by

cosθ̂|r|θ,ϕ=r̂zr̂|r|θ,ϕ=eiϕL̂zr̂zeiθL̂y1r̂|0,0,r=eiϕL̂zeiθL̂yeiθL̂yr̂zeiθL̂ybraiding1r̂|0,0,r=eiϕL̂zeiθL̂ysinθr̂x+cosθr̂zr̂|0,0,r=cosθ|r|θ,ϕ.
(41)

We can accomplish this because the eigenvalue–eigenvector relationship given above is unchanged when we change the value of r in the radial ket. Hence, the operator cosθ̂ acts as the identity on the |r⟩ ket and can be taken to act solely on the |θ, ϕ⟩ ket, yielding cosθ̂|θ,ϕ=cosθ|θ,ϕ. We can proceed similarly to verify that this separation holds true for the three other trigonometric operators. The argument for the radial operator acting only on |r⟩ can also be easily verified.

We can then take the limit δ → 0+ so that expiδp̂1. Gathering the final results together, we have established that

|r=eirp̂rir̂|r=0|θ,ϕ
(42)

and

r|=θ,ϕ|r=0|eirp̂r+ir̂.
(43)

These final expressions are a simplification of the translation operator in spherical coordinates when it acts on the state at the origin.

In order to calculate an energy wavefunction in position space, we must also decompose the energy eigenstate vector of the Hamiltonian according to a radial and angular momentum tensor-product state via

|Ψ=|ψr|l,m,
(44)

where the eigenvalues of L̂2 and L̂z (when acting on the state |l, m⟩) are l(l + 1)ℏ2 and mℏ, respectively. It follows that we can write the wavefunction as the product of the radial wavefunction and the angular momentum eigenfunction according to

r|Ψ=r|ψrθ,ϕ|l,m
(45)

and

Ψ(r)=ψr(r)Yl,m(θ,ϕ),
(46)

where

Yl,m(θ,ϕ)=θ,ϕ|l,m
(47)

denotes the standard spherical harmonic. A derivation of the spherical harmonic in a representation-independent way using the rotation operators of the “north-pole” state can be found in Ref. 7. The radial wavefunction is thus given by

ψr(r)=r|ψr=r=0|eirpr+ir̂|ψr.
(48)

This is the relation that we will use to determine the radial wavefunctions for the three-dimensional hydrogen atom.

We now turn to the expression of the two-dimensional translation operator in plane polar coordinates. In terms of Cartesian coordinates, we have

T̂(ρ)=eiρp̂=eirxp̂x+ryp̂y,
(49)

with

ρ=exrx+eyry,p̂=exp̂x+eyp̂y.
(50)

We use |0ρ as the position-space eigenvector at the origin for this two-dimensional case. We thus have

|ρ=T̂(ρ)|0ρ.
(51)

Probably the simplest way to go from |0ρ to |ρ is to translate by ρ in the +x-direction and then rotate by ϕ about the +z-direction. Indeed, this is the usual way of defining plane polar coordinates, ρ and ϕ. The corresponding operator is

T̂(ρ)=eiϕL̂zeiρp̂xeiϕL̂z.
(52)

To show that this is correct, we use the braiding relation to obtain

T̂(ρ)=eiρ(p̂xcosϕ+p̂ysinϕ).
(53)

Since

rx=ρcosϕandry=ρsinϕ,
(54)

we immediately see that Eq. (53) reduces to Eq. (49).

From Eqs. (A22) and (A23), we obtain

p̂x=p̂ρi2ρ̂cosϕ̂p̂ϕsinϕ̂
(55)

and

p̂y=p̂ρi2ρ̂sinϕ̂+p̂ϕcosϕ̂.
(56)

We use Eqs. (52) and (55) to write

T̂(ρ)=eiϕL̂zeiρp̂ρi2ρ̂cosϕ̂p̂ϕsinϕ̂eiϕL̂z.
(57)

We note that this is an operator identity.

A second way to express T̂(ρ) in plane polar coordinates is simply to use Eqs. (54)–(56) to substitute for the Cartesian quantities in Eq. (49). In this way, we obtain

rxp̂x+ryp̂y=ρp̂ri2ρ̂(cosϕcosϕ̂+sinϕsinϕ̂)p̂ϕ(cosϕsinϕ̂sinϕcosϕ̂),
(58)
rxp̂x+ryp̂y=ρp̂ri2ρ̂cos(ϕ̂ϕ)p̂ϕsin(ϕ̂ϕ),
(59)

and

rxp̂x+ryp̂y=eiϕL̂zρp̂ri2ρ̂cosϕ̂p̂ϕsinϕ̂eiϕL̂z.
(60)

Upon substitution of this into Eq. (49), we immediately obtain Eq. (57).

Next, we determine how the general operator form of the translation operator in plane polar coordinates simplifies when it acts on the origin state in two dimensions, |0ρ, similar to what we did in the three-dimensional case.

We introduce the same pair of δ-dependent exponentials to the left of T̂(ρ), where δ is a two-dimensional vector along ρ, move the rightmost exponential factor to the right through T̂(ρ), and operate on |0ρ. We thus obtain

|ρ=eiδp̂T̂(ρ)|δx,δy.
(61)

The choice that δ lies along the same direction as ρ implies that Eq. (29) holds and

eiϕL̂z|δx,δy=|δ,0,
(62)

where δ is given by Eq. (33) with δz = 0. Note that

cosϕ̂|δ,0=r̂xρ̂|δ,0=|δ,0
(63)

and

sinϕ̂|δ,0=r̂yρ̂|δ,0=0.
(64)

We expand the exponential containing p̂ρ in Eq. (57) and use the action of cosϕ̂ and sinϕ̂ on |δ, 0⟩ to simplify each term and then re-sum to obtain expiρp̂ρi2ρ̂ acting on the ket. This yields

|ρ=eiδp̂eiϕL̂zeiρp̂ρi2ρ̂|δ,0.
(65)

Since L̂z commutes with p̂ρ and ρ̂, we can separate the radial and angular degrees of freedom according to

|δx,δy=eiϕL̂z|δ,0=|ρ=δ|ϕ.
(66)

We define the radial coordinate eigenstate to satisfy

ρ̂|ρ=ρ|ρ.
(67)

A direct calculation gives ρ̂|δx,δy=δ|δx,δy justifying the label ρ = δ in Eq. (66). We similarly define the angular state, |ϕ⟩, to be the state that satisfies

cosϕ̂|ϕ=cosϕ|ϕ,sinϕ̂|ϕ=sinϕ|ϕ.
(68)

Note that the operators r̂x and r̂y cannot operate solely on the states |ρ⟩ or |ϕ⟩; that is, the domain of those operators lies outside of the space of the eigenstates of the radial-position operator ρ̂ and of the angle ket |ϕ⟩. Of course, they can act on the tensor-product state |ρ⟩ ⊗ |ϕ⟩ by decomposing the Cartesian position operators into their polar coordinate counterparts.

However, the operator ρ̂ does act only on |ρ⟩, which is its eigenstate, and it acts as the identity operator on |ϕ⟩. Similarly, cosϕ̂ and sinϕ̂ act nontrivially on |ϕ⟩, but as the identity on |ρ⟩, as derived below. Consider cosϕ̂ acting on the state |ρ cos ϕ, ρ sin ϕ⟩ = |ρ⟩ ⊗ |ϕ⟩, which is given by

cosϕ̂|ρ|ϕ=r̂xρ̂|ρ|ϕ=r̂xeiϕL̂z1ρ̂|ρ,0=eiϕL̂zeiϕL̂zr̂xeiϕL̂zbraiding1ρ̂|ρ,0=eiϕL̂zcosϕr̂xsinϕr̂yρ̂|ρ,0=cosϕ|ρ|ϕ.
(69)

Note that the eigenvalue–eigenvector relationship given above is unchanged when we change the value of ρ in the radial ket. Hence, the operator cosϕ̂ acts as the identity on the |ρ⟩ ket and can be taken to act solely on the |ϕ⟩ ket, yielding cosϕ̂|ϕ=cosϕ|ϕ. Similarly, this separation also holds for sinϕ̂. The argument for the radial operator acting only on |ρ⟩ can be verified by a similar argument.

We now take the limit δ → 0+ so that expiδp̂1. Gathering the final results together, we have established that

|ρ=eiρp̂ρi2ρ̂|ρ=0|ϕ
(70)

and

ρ|=ϕ|ρ=0|eiρp̂ρ+i2ρ̂.
(71)

These final expressions are the simplification of the translation operator in plane polar coordinates that we sought when it acts on the state at the origin.

To calculate an energy wavefunction in position space, we decompose the energy eigenstate of the Hamiltonian into the tensor product of a radial state and an eigenstate of L̂z. This is given by

|Ψ=|ψρ|m,
(72)

with L̂z|m=m|m. The wavefunction is then expressed as the product of a radial wavefunction and a z-component of angular momentum eigenfunction according to

ρ|Ψ=ρ|ψρϕ|m
(73)

or

Ψ(ρ)=ψρ(ρ)Ym(ϕ).
(74)

Here,

Ym(ϕ)=ϕ|m=ϕ=0|eiL̂z|m=ϕ=0|meimϕ=eimϕ2π
(75)

denotes the standard eigenfunction of L̂z. We used the fact that normalization requires ϕ=0|m=1/2π. The radial wavefunction is then given by

ψρ(r)=ρ|ψρ=ρ=0|eiρp̂ρ+i2ρ̂|ψρ.
(76)

This is the relation that we use to determine the radial wavefunctions for the two-dimensional hydrogen atom in Sec. III.

We now show how these translation operators can be used to calculate wavefunctions of the Coulomb problem in three and two dimensions. We start in three dimensions, where we have

Ĥ3d=p̂x2+p̂y2+p̂z22μe2r̂.
(77)

Here, μ is the reduced mass for the atom and ∓e are the electronic and nuclear charges, respectively. Using the decomposition of the kinetic energy into radial and angular degrees of freedom, we find that the Hamiltonian can be re-written as

Ĥ3d=p̂r22μ+L̂L̂2μr̂2e2r̂.
(78)

We use a tensor product to represent the state vector in terms of the angular momentum eigenstate, |l, m⟩, and the radial state |ψr⟩ via |ψ⟩ = |ψr⟩ ⊗ |l, m⟩. Then, because L̂L̂|l,m=2l(l+1)|l,m, we have

Ĥ3d|ψr|l,m=p̂r22μ+2l(l+1)2μr̂2e2r̂|ψr|l,m=Ĥl3d|ψr|l,m,
(79)

which defines the Hamiltonian Ĥl3d for a specific angular momentum sector. This Hamiltonian acts only on the radial part of the state vector because all terms in Ĥl3d commute with L̂.

The Schrödinger factorization method is used to factorize each of these angular-momentum-sector Hamiltonians via

Ĥl3d=p̂r22μ+2l(l+1)2μr̂2e2r̂=B̂lB̂l+El,
(80)

where the correct lowering operator satisfies

B̂l=12μp̂ri1(l+1)a0l+1r̂,
(81)

which we can verify with the use of the commutation relation [p̂r,r̂k]=ikr̂k1 (which can be derived by induction from the radial momentum canonical commutation relation using the techniques in the  Appendix). Note that the reason one can see for those two specific terms in B̂l is that when we square the term in parentheses and when we commute it with p̂r, the resulting terms are always a constant term, a linear term in 1/r̂, and a quadratic term (1/r̂2). By adjusting the coefficients, we can arrange for the product B̂lB̂l to have the required form. The symbol a0=2μe2 is the Bohr radius, and El=e22(l+1)2a0.

We denote the eigenstate of Ĥl3d, corresponding to eigenvalue El=n−1 as |n, l = n − 1⟩. Here, n, which is a positive integer, is the standard principal quantum number, and the energy is degenerate for all l such that 0 ≤ ln − 1. We then have

En1=e22n2a0,
(82)

which, unfortunately, is conventionally denoted as En. To avoid confusion in the equations that follow, we do not redefine it, as is customarily done. Further note that the choice for the lowering operator in Eq. (81) was made so that the set of energies En−1 form an increasing sequence for 1 ≤ n < . This choice for the increasing sequence guarantees that the eigenfunctions are all normalizable bound states.

Since B̂lB̂l is a non-negative semidefinite operator, it follows from Eq. (80) that the ground state |n, n − 1⟩ of Ĥn13d satisfies

Ĥn13d|n,n1=En1|n,n1,
(83)

with

B̂n1|n,n1=0.
(84)

As we will see, this condition, which we term the subsidiary condition, facilitates the determination of all the eigenstates |n, l⟩.

One can also directly verify that when the raising and lowering operators act in the “wrong” order, we have

B̂lB̂l=p̂r22μ+2(l+1)(l+2)2μr̂2e2r̂El=Ĥl+13dEl.
(85)

This allows us to derive the intertwining relationship when a B̂l operator is moved to the left past a Ĥl3d operator. In particular, we have

Ĥl3dB̂l=B̂lB̂l+ElB̂l=B̂lB̂lB̂l+ElB̂l=B̂lB̂lB̂l+El=B̂lĤl+13d.
(86)

Hence, when a B̂l operator starts on the right and moves to the left through Ĥl3d, it shifts the index l of the Hamiltonian upward by one.

We use this intertwining identity to construct all the energy eigenstates that have energy En−1. Note that this approach is different from the original Schrödinger approach. It is the simplest way to establish all the energy eigenstates. For each n, we can find a total of n states with different total angular momenta that are eigenstates. The key observation is that the following set of states are eigenstates:

|n,l=CnlB̂lB̂l+1B̂n3B̂n2|n,n1,
(87)

where 0 ≤ ln − 1: note that when l = n − 1, there are noB̂ operators needed and when l = 0, the string runs from l = n − 2 down to l = 0. In particular, there is noB̂1 operator because El → − as l → −1. This terminates the chain at l = 0. The number Cnl is a normalization constant, which we determine below. To prove that |n, l⟩ is an eigenstate of energy En−1, we simply operate with Ĥl3d on |n, l⟩. Using the intertwining relation, to move the H3d operator to the right, increasing its index by one with every step, we find that

Ĥl3d|n,l=Ĥl3dCnlB̂lB̂l+1B̂n3B̂n2|n,n1=CnlB̂lĤl+13dB̂l+1B̂n3B̂n2|n,n1=CnlB̂lB̂l+1B̂n3Ĥn23dB̂n2|n,n1=CnlB̂lB̂l+1B̂n3B̂n2Ĥn13d|n,n1=e22n2a0|n,l
(88)

since the state |n, n − 1⟩ is an eigenstate of Ĥn13d with eigenvalue −e2/2n2a0. We see that for a given n, all the states |n, l⟩, with 0 ≤ ln − 1, belong to the eigenvalue En−1. Hence, we have constructed eigenstates of each of the lth angular-momentum-sector Hamiltonians with 0 ≤ ln − 1. When we make a tensor product of such a state with an |l, m⟩ angular momentum state, it becomes an eigenstate of the full Coulomb Hamiltonian.

Before further moving in the derivation, we illustrate schematically what the energy levels are and how the different eigenstates interrelate. This is depicted in Fig. 2.

FIG. 2.

Plot of the energy levels of the three-dimensional Coulomb problem up to n = 4. The states |n, n − 1⟩ are the rightmost states in each row. Each additional B̂ operator moves us one step to the left (there are no more B̂ operators to apply after l = 0). All energies in a given row are degenerate. The dashed line shows where E = 0.

FIG. 2.

Plot of the energy levels of the three-dimensional Coulomb problem up to n = 4. The states |n, n − 1⟩ are the rightmost states in each row. Each additional B̂ operator moves us one step to the left (there are no more B̂ operators to apply after l = 0). All energies in a given row are degenerate. The dashed line shows where E = 0.

Close modal

We also use the intertwining relation to normalize the state and determine Cnl. This is done by simply computing the norm

1=n,l|n,l=|Cnl|2n,n1|B̂n2B̂lB̂lB̂n2|n,n1.
(89)

We start with the innermost pair of raising and lowering operators and note that B̂lB̂l=Ĥl+13dEl. If we move this operator through to the right, increasing its index by one with each step, until it acts directly on the state |n, n − 1⟩, we see that the intertwining relation will convert the Hamiltonian to the one corresponding to l = n − 1. This can be immediately evaluated against the state |n, n − 1⟩, yielding the eigenvalue En−1. The net effect is we remove the product of the two operators B̂lB̂l and replace it with the factor (En−1El). Repeating nl − 2 more times, we find that

Cnl=1k=ln2(En1Ek)
(90)
=2a0n2e2nl1(n+l)![(n1)!]2(2n1)!(nl1)!(l!)2.
(91)

Note that this final result requires that the initial state |n, n − 1⟩ be normalized, i.e., ⟨n, n − 1|n, n − 1⟩ = 1. We will ensure that this is true.

The next step is to show that the string of B̂ operators acting on |n, n − 1⟩ is proportional to a Laguerre polynomial in r̂ (multiplied by a power of 1/r̂) acting on the same state. To do this, we need to go through a proof by induction. Our starting point is to observe that the subsidiary condition in Eq. (84) can be rewritten as

p̂r|n,n1=i1na0nr̂|n,n1.
(92)

This allows us to replace the radial momentum operator acting on the state |n, n − 1⟩ by the sum of a constant and a term that goes like 1/r̂ (acting on the same state). Given that the commutation relation of the radial momentum with an inverse power of r̂ increases the power by one as well, one can immediately see that the string of B̂ operators acting on |n, n − 1⟩ will be a polynomial of degree nl − 1 in 1/r̂. By factoring out 1/r̂nl1, we are left with a polynomial of degree nl − 1 in r̂. Recognizing that the dimensionality of a B̂ operator is /(μa0) and using 2r̂/na0 as the expansion parameter for the polynomial, we have that

B̂lB̂n2|n,n1=2i2μna0nl1na02r̂nl1j=0nl1bj(l)2r̂na0j|n,n1.
(93)

Note that we are suppressing a label of n corresponding to the principal quantum number in the notation for the b coefficients to streamline the notation; one must remember that a set of b coefficients is generated for each principal quantum number n. We choose the base case to correspond to l = n − 2, where the polynomial is just equal to

B̂n2|n,n1=i2μ2n1n(n1)a02n1r̂|n,n1=2i2μna0na02r̂2n+1+2n12(n1)2r̂na0|n,n1,
(94)

so that b0(n2)=2n+1 and b1(n2)=(2n1)/2(n1). (One could have taken the base case as l = n − 1, with b0(n1)=1, but we chose not to because that is a trivial base case.) To evaluate the induction step, we remove the leftmost B̂l and collect what remains in terms of the polynomial for l + 1. Then, we operate the p̂r operator to the right, commuting it through the r̂ operators until it can act on the state |n, n − 1⟩, where we can use Eq. (92) to replace it by a binomial in 1/r̂. After collecting the coefficients of powers of r̂, we find that

j=0nl1bj(l)2r̂na0j=j=0nl12r̂na0j(j+2l+3)bj(l+1)+n+l+12(l+1)bj1(l+1),
(95)

where we set b1(l+1)=bnl1(l+1)=0. The key to determining that these results yield associated Laguerre polynomials is to compute the ratio of successive coefficients of the lth polynomial using the two formulas we have. This gives

bj+1(l)bj(l)=(j+2l+4)bj+1(l+1)bj(l+1)n+l+12(l+1)(j+2l+3)n+l+12(l+1)bj(l+1)bj1(l+1)1for0jnl2.
(96)

Here, again, we note that b1(l+1)=bnl1(l+1)=0. This brings us to the induction hypothesis. The ratio is given by

bj+1(l)bj(l)=jn+l+1(j+1)(j+2l+2).
(97)

One can immediately see from Eq. (94), for l = n − 2, that the base case gives b1(n2)/b0(n2)=1/2(n1), which agrees with the induction hypothesis in Eq. (97) for l = n − 2 and j = 0. Hence, we assume that it holds for l + 1 and substitute into Eq. (96) to show it holds for l. This yields

bj+1(l)bj(l)=(jn+l+2)(j+1)(n+l+1)2(l+1)(j+2l+3)1(n+l+1)j2(l+1)(jn+l+1).
(98)

Putting everything over common denominators yields

bj+1(l)bj(l)=(jn+l+1)2(l+1)(jn+l+2)(j+1)(n+l+1)(j+2l+3)(j+1)2(l+1)(jn+l+1)j(n+l+1).
(99)

Expanding the terms in the numerator and denominator in the square brackets, factorizing the expansion, and simplifying then reduces this result to the one in Eq. (97), which establishes the induction proof. It turns out that the coefficients of a polynomial that satisfy Eq. (97) are associated Laguerre functions. We use the definition of the associated Laguerre function given by Powell and Crasemann in their quantum mechanics textbook,8 

Lkα(x)=j=0m(1)jj!k+αkjxj=j=0kaj(α,k)xj,
(100)

which defines the coefficients of the polynomial aj(α,k). We immediately see from Eq. (100) that

aj+1(α,k)aj(α,k)=jk(j+1)(α+j+1).
(101)

This then tells us that the polynomial given by the string of B̂ operators acting on |n, n − 1⟩ is proportional to the associated Laguerre function with argument given by 2r̂/na0, α = 2l + 1, and k = nl − 1. To find the constant of proportionality [in addition to the power (2r̂/na0)nl1, which we already knew from Eq. (93)], we evaluate the constant term (no r̂ dependence or the maximal power of r̂ in the sum) in Eq. (93). It is given by

2i2μna0nl1bnl1(l)=i2μnl1k=l+1n11na0+1ka0,
(102)

where the right-hand side is found from the constant terms in each B̂ and from each p̂r in each B̂ when it acts on |n, n − 1⟩. This constant term is not modified by the commutators of the radial momentum with inverse powers of r̂. From this, we learn that

bnl+1(l)=12nl1l!(2n1)!(n1)!(n+l)!.
(103)

However, if we instead express bnl1(l) in terms of the maximal coefficient of the Laguerre polynomial, we find that

bnl+1(l)=Canl1(2l+1,nl1)=C(1)nl1(nl1)!,
(104)

where C′ is the proportionality constant we need to determine. We immediately learn that

C=12nl1l!(nl1)!(2n1)!(n+l)!(n1)!.
(105)

Putting this all together, we have derived the identity that

B̂lB̂l+1B̂n3B̂n2|n,n1=i2μna0nl1l!(nl1)!(2n1)!(n+l)!(n1)!×na02r̂nl1Lnl12l+12r̂na0|n,n1.
(106)

Finally, we multiply by Cnl from Eq. (91) and find that

|n,l=(i)nl1(nl1)!(2n1)!(n+l)!na02r̂nl1Lnl12l+12r̂na0|n,n1.
(107)

Since the eigenvector is defined only up to a complex phase, we choose to remove the factor of (−i)nl−1 in the following.

We are now ready to compute the wavefunction. We take the overlap of the position eigenstate with the energy eigenstate or

(θϕ|r|)(|n,l|l,m).
(108)

Using the fact that7 

Ylm=θϕ|lm,
(109)

we find that

ψnlm(r,θ,ϕ)=(nl1)!(2n1)!(n+l)!r|na02r̂nl1Lnl12l+12r̂na0|n,n1Ylm(θ,ϕ).
(110)

Because r̂|r=r|r, we immediately find that

ψnlm(r,θ,ϕ)=(nl1)!(2n1)!(n+l)!r|n,n1na02rnl1Lnl12l+12rna0Ylm(θ,ϕ).
(111)

What remains is to determine the first wavefunction in the chain, ϕn(r) = ⟨r|n, n − 1⟩. This is where we need to use the translation operator in spherical coordinates. The radial translation operator is an exponential of irp̂r+ir̂/ when acting on the position eigenvector at the origin. While one can evaluate the exponential of an operator acting on a state via expanding the power series term by term, a more efficient evaluation can be accomplished if one evaluates the operator acting on one of its eigenvectors—then, the exponentiation becomes trivial. This is the strategy we adopt here.

We start with some additional operator identities. First, note that [p̂r,r̂n+1]=i(n1)/r̂n. (See the  Appendix for details.) Hence, we use Eq. (92) and the commutator to find that

p̂r1r̂n1|n,n1=i1na0r̂n11r̂n|n,n1,
(112)

or

p̂r+ir̂1r̂n1|n,n1=ina01r̂n1|n,n1.
(113)

Note that this says that the state (1/r̂n1)|n,n1 is an eigenvector for the operator p̂r+ir̂ with eigenvalue iℏ/na0. An imaginary eigenvalue is possible because this operator is not Hermitian.

We are now ready to use the translation operator in spherical coordinates. Since θϕ|=0,0,1|exp(iθL̂y/)exp(iϕL̂z/), we have that

r|=0r|expirp̂r+ir̂.
(114)

The wavefunction ϕn(r) then becomes

ϕn(r)=r|n,n1=r|r̂n1r̂n1|n,n1=rn1r|1r̂n1|n,n1=rn10r|expirp̂r+ir̂1r̂n1|n,n1.
(115)

Because the state on the right is an eigenvector for p̂r+ir̂, we immediately find that

ϕn(r)=rn1exprna00r|1r̂n1|n,n1.
(116)

Note that the term 0r|1r̂n1|n,n1 is a constant because Eq. (116) shows that

limr01rn1ϕn(r)=0r|1r̂n1|n,n1.
(117)

Its precise value is determined by normalization. We find that

1=0r|1r̂n1|n,n120drr2ne2rna0,
(118)

which yields

ϕn(r)=2na0n+121(2n)!rn1exprna0.
(119)

While this result can be calculated in many different ways, one cannot calculate this wavefunction using the translation operator in Cartesian coordinates; the translation operator must be expressed in terms of spherical coordinates because it involves a power series in r that includes odd powers.

We can now summarize our final wavefunction by combining all our results together,

ψnlm(r,θ,ϕ)=2na032(nl1)!2n(n+l)!2rna0lLnl12l+12rna0erna0Ylm(θ,ϕ),
(120)

which is the standard result for the three-dimensional Coulomb bound-state wavefunctions using the Laguerre polynomial definition in Eq. (100). We want to stress that the entire calculation was based on the representation-independent approach described in the introduction. All the steps in the derivation used operator algebra. We never needed to represent momentum operators in terms of spatial derivatives. It is comforting to know that wavefunctions can be calculated in such a representation-independent way.

Having completed our first example, we now move on to the second. We will expedite the description because many of the techniques for the two-dimensional case are similar to those of the three-dimensional case. However, the two-dimensional case does illustrate some interesting new twists, so it is important to describe it carefully.

The Coulomb Hamiltonian in two dimensions is given by

Ĥ2d=p̂x2+p̂y22μe2ρ̂.
(121)

The kinetic energy can again be decomposed into radial and angular components (see the  Appendix),

H2d=p̂ρ22μ+L̂z2242μρ̂2e2ρ̂,
(122)

with

p̂ρ=1ρ̂r̂xp̂x+r̂yp̂yi2.
(123)

Note that in two dimensions, we do have a quantum correction to the kinetic energy (given by the 2/8μρ̂2 term). Here, the angular momentum states are given by eigenstates of L̂z, which satisfy L̂z|m=m|m. Therefore, we form the energy eigenstates as a tensor product of radial and angular momentum states via |ψ⟩ = |ψρ⟩ ⊗ |m⟩. Operating the Hamiltonian onto the tensor-product state yields

H2d|ψρ|m=p̂ρ22μ+2(m214)2μρ̂2e2ρ̂|ψρ|m=Hm2d|ψρ|m,
(124)

where the second line defines the set of Hamiltonians, Hm2d, that operate only on the radial state |ψρ⟩. Note here that the index m can be a positive or negative integer, unlike in the three-dimensional case, where l is a non-negative integer. Since each of these Hamiltonians depends on m only through m2, we immediately learn that the radial eigenfunctions and the energy eigenvalues depend only on the magnitude of m, that is, on |m|.

We continue to use the Schrödinger factorization method for m ≥ 0 via

Hm2d=p̂ρ22μ+2m2142μρ̂2e2ρ̂=B̂mB̂m+Em.
(125)

A quick calculation tells us that

B̂m=12μp̂ρi1m+12a0m+12ρ̂
(126)

and

Em=e22a0m+122.
(127)

Note that for m ≥ 0, the energies form an increasing sequence, which is required for all the energy eigenstates to be normalizable. While the algebra above holds for all m, we first focus on working with non-negative m; we will describe how to handle negative m later. In particular, we already know that the radial part of the eigenvectors will be identical, as will the energies, so we will not be using the above formulas with negative m to solve these problems.

Computing the product of the two operators in opposite order gives us

B̂mB̂m+Em=p̂ρ22μ+2(m+1)2142μρ̂2e2ρ̂=Hm+12d.
(128)

This relation is essentially the same relation as we had in three-dimensions, Eq. (85), so that we immediately find the corresponding intertwining relation (for m ≥ 0)

Hm2dB̂m=B̂mHm+12d,
(129)

which can be employed to find eigenstates in the same fashion: we have the eigenvectors given by

|n,m=CnmB̂mB̂m+1B̂n3B̂n2|n,n1,
(130)

for m ≥ 0, with the eigenvalues equal to

En1=e22a0n122
(131)

and Cnm being a normalization constant. The state |n, m = n − 1⟩ = |n, n − 1⟩ satisfies

B̂n1|n,n1=0.
(132)

As seen in the three-dimensional case, the normalization constant (for m ≥ 0) is given by

Cnm=1k=mn2(En1Ek)=a0n122enm1(n+m1)![(2n3)]2(2n2)!(nm1)![(2m1)]2.
(133)

We now discuss some interesting observations about what happens when m < 0. For example, one can see from Eq. (126) that B̂|m|=B̂|m|1. Hence, if we extend the eigenstates in Eq. (130) for m < 0, then, because we have that

B̂|m|B̂1B̂0B̂|m|1=B̂|m|1B̂0B̂0B̂|m|1=B̂|m|1B̂1(H12dE0)B̂1B̂|m|1=B̂|m|1B̂1B̂1B̂|m|1(H|m|2dE0)=(H|m|2dE|m|1)(H|m|2dE0),
(134)

we can show the equivalence of the radial component of the eigenvectors for negative m with positive m. If we let the product of operators in Eq. (134) act on |n, |m|⟩, then each H|m|2d yields E|m|, so we find that

|n,|m|=|n,|m|
(135)

directly from the operator identities. This confirms our statement before that the radial component of the eigenvector for −|m| is identical to the eigenvector for |m| (with the same principal quantum number n). Further note that the product of numerical factors is precisely the numerical factors to guarantee that Cn−|m| = Cn|m|, which is why we have an equality in Eq. (135) rather than a proportionality.

At this stage, we show the spectra along with the operator representation of the radial components of the eigenvectors in Fig. 3. All the states in a horizontal row (fixed n, −n + 1 ≤ mn − 1) are degenerate with energy En−1. The column with m = 0 has the largest number of eigenstates. As |m| increases, the number of total eigenstates decreases by 1 for each step moving further away from m = 0 (to the right or to the left).

FIG. 3.

Plot of the energy levels of the two-dimensional Coulomb problem up to n = 4. The states |n, n − 1⟩ are the rightmost states in each row. Each additional B̂ operator moves us one step to the left (there are no more B̂ operators to apply after m = −n + 1). All energies in a given row are degenerate. The dashed line shows where E = 0.

FIG. 3.

Plot of the energy levels of the two-dimensional Coulomb problem up to n = 4. The states |n, n − 1⟩ are the rightmost states in each row. Each additional B̂ operator moves us one step to the left (there are no more B̂ operators to apply after m = −n + 1). All energies in a given row are degenerate. The dashed line shows where E = 0.

Close modal

Our next goal is to find the wavefunctions in position space, and we now assume m ≥ 0 again. As mentioned before, we see that the string of B̂ operators acting on the state |n, n − 1⟩ will create a polynomial of degree nm − 1 in 1/ρ̂. If we factor out the term 1/ρ̂nm1, then the polynomial is an order nm − 1 polynomial in ρ̂. Just as we saw happen in three dimensions, it turns out to be a Laguerre polynomial for the two-dimensional case too.

To see this, we start with a definition of the coefficients of the nth polynomial (for m ≥ 0, without loss of generality),

B̂mB̂n2|n,n1=2i2μn12a0nm1n12a02ρ̂nm1j=0nm1bj(m)2ρ̂n12a0j|n,n1.
(136)

Again, we suppress the index n in the labeling of the b coefficients for notational simplicity. We consider the base case first, corresponding to m = n − 2. First, we need to determine a modification of the subsidiary condition in Eq. (132) to determine what happens when the radial momentum acts on |n, n − 1⟩,

p̂ρ|n,n1=i1n12a0n12ρ̂|n,n1.
(137)

Next, we have that the base case is

B̂n2|n,n1=i2μ2n2n12n32a02n2ρ̂|n,n1=2i2μn12a0n12a02ρ̂(2n2)+2n22n32ρ̂n12a0|n,n1
(138)

so that b0(n2)=(2n2) and b1(n2)=(2n2)/(2n3). This means that b1(n2)/b0(n2)=1/(2n3), a fact we shall use momentarily.

After establishing the base case for the proof by induction, we next examine the m-th polynomial by splitting off the operator B̂m and having it act on the (m + 1)st polynomial. This yields

j=0nm1(j+2m+2)bj(m+1)+m+n2m+1bj1(m+1)2ρ̂(n12)a0j=j=0nm1bj(m)2ρ̂n12a0j.
(139)

Then, we determine the ratio of subsequent coefficients of the polynomials, just like before, and find that

bj+1(m)bj(m)=(j+2m+3)bj+1(m+1)bj(m+1)m+n2m+1(j+2m+2)m+n2m+1bj(m+1)bj1(m+1)1for0jnm2,
(140)

where we again use b1(m+1)=bnm1(m+1)=0.

The induction hypothesis is that

bj+1(m)bj(m)=jn+m+1(j+1)(j+2m+1).
(141)

Recalling the base case, with m = n − 2 and j = 0, we have that the ratio equals −1/(2n − 3), which is what we already calculated above for the base case. Substituting the induction hypothesis for m + 1 (evaluated for j and for j − 1) into Eq. (140) yields

bj+1(m)bj(m)=(jn+m+1)[(2m+1)(jn+m+2)(m+n)(j+1)](j+1)(j+2m+2)[(2m+1)(jn+m+1)(m+n)j].
(142)

Expanding the terms in the square brackets, factorizing them, and simplifying produces the results in Eq. (141), which completes the induction proof. Given the ratio of the consecutive coefficients in the Laguerre polynomial in Eq. (101), we see that the string of B operators acting on |n, n − 1⟩ yields the Laguerre polynomial (with α = 2m and k = nm − 1) multiplied by a power of ρ̂ multiplied by a constant, or

B̂mB̂n2|n,n1=C2i2μn12a0nm1n12a02ρ̂nm1Lnm12m2ρ̂n12a0|n,n1.
(143)

Using the same approach as described before, we compute the coefficient of the term that has no ρ̂ dependence. We express the result in terms of the b coefficients and equate it to the constant term found from using the subsidiary condition when the string of B̂ operators acts on |n, n − 1⟩. We have

2i2μn12a0nm1bnm1(m)=i2μnm1k=mn21n12a0+1k+12a0,
(144)

which can be solved for bnm1(m) with

bnm1(m)=(2n2)!(2m1)(n+m1)!(2n3).
(145)

We also can calculate this coefficient from the Laguerre polynomial form. We have

bnm1(m)=Camn1(2m,nm1)=(1)nm1(nm1)!.
(146)

Hence, we have

C=(1)nm1(nm1)!(2n2)!(2m1)(n+m1)!(2n3)
(147)

and

|n,m=(i)nm1(nm1)!(2n2)!(n+m1)!n12a02ρ̂nm1Lnm12m2ρ̂n12a0|n,n1.
(148)

As mentioned before, we drop the irrelevant phase factor, (−i)nm−1. Now, we are ready to compute the wavefunctions. We need to determine the overlap of ⟨ϕ| ⊗ ⟨ρ| with |n, m⟩ ⊗ |m⟩. Since ϕ|m=exp(imϕ)/2π and ρ̂|ρ=ρ|ρ, we find that

ψnm(ρ)=(nm1)!(2n2)!(n+m1)!ρ|n,n1n12a02ρnm1Lnm12m2ρn12a0.
(149)

This leaves us with the task of computing the wavefunction for maximal m with a given n, that is, with m = n − 1.

Here is where we need to use the translation operator in polar coordinates. We derived the translation operator when acting on the 0ρ| bra in Eq. (76). Hence, we have that

ϕn(ρ)=ρ|n,n1=0ρ|expiρp̂ρ+i2ρ̂|n,n1.
(150)

As mentioned before, the best way to evaluate this expression is to find the eigenvector of p̂ρ+i2ρ̂. Using Eq. (137), we see that

p̂ρ1ρ̂n1|n,n1=i1n12a012ρ̂1ρ̂n1|n,n1,
(151)

which leads to the eigenvalue/eigenvector relationship

p̂ρ+i2ρ̂1ρ̂n1|n,n1=in12a01ρ̂n1|n,n1.
(152)

Hence, we have that

ϕn(ρ)=0ρ|1ρ̂n1|n,n1ρn1expρn12a0.
(153)

Normalizing gives us

0ρ|1ρ̂n1|n,n1=2n12a0n1(2n1)!.
(154)

Summarizing, we find that the normalized wavefunction is

ψnm(ρ,ϕ)=2n12a0|m|+1(n|m|1)!(2n1)(n+|m|1)!ρ|m|expρn12a0Ln|m|12|m|2ρn12a0eimϕ2π.
(155)

This completes the calculation of the wavefunctions of the two-dimensional Coulomb problem.

In this paper, we have shown how one can calculate wavefunctions in a representation-independent manner. The strategy starts from the origin in position space, translates to the position r, and then computes the inner product with the energy eigenstate. Doing this for some central-potential problems requires us to convert the translation operator from Cartesian coordinates to spherical or polar coordinates because a Taylor series expansion for the wavefunctions of some problems does not exist in terms of the Cartesian coordinates. (This is true, for example, in the Coulomb problem but is not true for the two- or three-dimensional simple harmonic oscillator.) We illustrated how to convert from Cartesian coordinates to spherical (or polar) coordinates and then illustrated how the operator expression simplifies when it acts on the position-space eigenvector at the origin. Finally, we showed how such a procedure works by solving for the wavefunctions of the Coulomb problem in three and two dimensions. To do this requires us to use Schrödinger’s factorization method, but we do so at the operator level only and do not convert the operators into the position-space representation. We want to emphasize that performing calculations in a representation-independent fashion illustrates that essentially all quantum properties are derived from the existence of a few eigenvectors (such as the position-space eigenvector at the origin and the ground-state eigenvector of the Coulomb problems) and the canonical commutation relations, [r̂α,p̂β]=iδαβ. No other assumptions are needed. This methodology is quite general, but it is simpler for position-space wavefunctions than for momentum-space wavefunctions. This is because the generic factorization method has raising and lowering operators that are linear in momentum operators but often are nonlinear in the position operators. Hence, it is easier to determine how the momentum operator acts on an energy eigenstate than to determine how a position operator acts. Nevertheless, this approach can be used in both position and momentum space and explicitly shows that these two wavefunctions are constructed in similar ways. The approach to determine them from “differential equations” is often quite different as the general formulation in momentum space is an integral equation rather than a differential equation, which is what is used in position space. This is true even though the two wavefunctions are directly related to each other via a Fourier transformation.

This work was supported by the National Science Foundation under Grant No. PHY-1915130. In addition, J.K.F. was supported by the McDevitt bequest at Georgetown University.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

In the early years of quantum mechanics, Pauli9 and Dirac10 showed how to use the canonical commutation relation to compute commutators of functions of r̂α and p̂α. We use the same method to compute the commutators needed in this work. To start, we compute the commutator of momentum with r̂2=r̂x2+r̂y2+r̂z2. Using the Leibniz rule for the commutator of a product, we immediately find that

p̂α,r̂2=r̂αp̂α,r̂α+p̂α,r̂αr̂α=2ir̂α.
(A1)

Note that we never use the Einstein summation convention in this paper. Similarly, defining r̂=r̂x2+r̂y2+r̂z2, the Leibniz rule shows that

p̂α,r̂2=r̂p̂α,r̂+p̂α,r̂r̂=2ir̂α
(A2)

after using the result in Eq. (A1). The commutator [p̂α,r̂] commutes with r̂ as described by Böhm,4 which follows from the triple commutator with r̂2,

r̂2,[p̂α,r̂]=r̂r̂,[p̂α,r̂]+r̂,[p̂α,r̂]r̂,
(A3)

and moving the r̂ operator into the second element of the outermost commutator to multiply the term [p̂α,r̂] for both nested commutators (which is valid because [r̂,r̂]=0) yields

r̂,r̂[p̂α,r̂]+[p̂α,r̂]r̂=r̂,p̂α,r̂2=r̂,(2ir̂α)=0
(A4)

after substituting Eq. (A1) into the innermost commutator. Because the square root of an operator is uniquely defined to have the same eigenvectors as the original operator (but all eigenvalues are equal to the positive square roots of the eigenvalues of the original operators), if an operator  commutes with another operator B̂, then the square root of  also commutes with B̂. Hence, from Eq. (A4), we have r̂,[p̂α,r̂]=0. Combining this with Eq. (A2) gives us

[p̂α,r̂]=ir̂αr̂.
(A5)

One uses the Leibniz rule again to compute the commutator with 1/r̂ via

0=[p̂α,1]=p̂α,r̂r̂=ir̂αr̂2+r̂p̂α,1r̂,
(A6)

which can be re-arranged to give

p̂α,1r̂=ir̂αr̂3.
(A7)

Similarly, using ρ̂=r̂x2+r̂y2, we find that [p̂α,ρ̂]=ir̂α/ρ̂ for α = x or y and it vanishes for α = z. We also find that [p̂α,1/ρ̂]=ir̂α/ρ̂3 with the same conditions on α.

Similarly, we define cosθ̂=r̂z/r̂, sinθ̂=ρ̂/r̂, cosϕ̂=r̂x/ρ̂, and sinϕ̂=r̂y/ρ̂. Then, using the Leibniz rule, we find that

p̂x,cosθ̂=ir̂xr̂zr̂3,p̂y,cosθ̂=ir̂yr̂zr̂3,p̂z,cosθ̂=iρ̂2r̂3,
(A8)
p̂x,sinθ̂=ir̂xr̂z2ρ̂r̂3,p̂y,sinθ̂=ir̂yr̂z2ρ̂r̂3,p̂z,sinθ̂=iρ̂r̂zr̂3,
(A9)
p̂x,cosϕ̂=ir̂y2ρ̂3,p̂y,cosϕ̂=ir̂xr̂yρ̂3,p̂z,cosϕ̂=0,
(A10)

and

p̂x,sinϕ̂=ir̂xr̂yρ̂3,p̂y,sinϕ̂=ir̂x2ρ̂3,p̂z,sinϕ̂=0.
(A11)

Of course, these “trigonometric function” operators commute with themselves and with ρ̂ and r̂ because all position operators commute with each other. Note that the angle operators are well defined only when they are arguments of the trigonometric functions. We never work with angle operators by themselves.

As described in the main text, we use the unit-vector operators (êr=sinθ̂cosϕ̂ex+sinθ̂sinϕ̂ey+cosθ̂ez, êθ=cosθ̂cosϕ̂ex+cosθ̂sinϕ̂eysinθ̂ez, and êϕ=sinϕ̂ex+cosϕ̂ey) to define the components of momentum in spherical coordinates; in spherical coordinates, the unit vectors must be operators, while in Cartesian space, they are not. Using the symmetric combination of the dot product of momentum with these unit vectors (given schematically as 12êp̂+12p̂ê) and after computing the “quantum corrections” (proportional to iℏ), we find that

p̂r=sinθ̂cosϕ̂p̂x+sinθ̂sinϕ̂p̂x+cosθ̂p̂zir̂,
(A12)
p̂θ=cosθ̂cosϕ̂p̂x+cosθ̂sinϕ̂p̂ysinθ̂p̂zicotθ̂2r̂,
(A13)

and

p̂ϕ=sinϕ̂p̂x+cosϕ̂p̂y.
(A14)

Note that p̂ϕ does not have a quantum correction due to reordering. This is because it is equal to L̂z/ρ̂.

Now that we have the components of momentum along the different spherical coordinate directions, we can use the commutation relations of the Cartesian components of momentum with the radial and angular operators to find that

p̂r,r̂=i,p̂r,ρ̂=isinθ̂
(A15)

and the commutator of p̂r with any trigonometric function of angles θ̂ and ϕ̂ is zero. We also have that p̂θ commutes with functions of r̂ and ϕ̂. However, we have

p̂θ,cosθ̂=isinθ̂r̂andp̂θ,sinθ̂=icosθ̂r̂.
(A16)

In addition, we have

p̂θ,ρ̂=icosθ̂.
(A17)

Finally, for p̂ϕ, we find that it commutes with trigonometric functions of θ̂ and arbitrary functions of r̂ and ρ̂. We also find that

p̂ϕ,cosϕ̂=isinϕ̂ρ̂andp̂ϕ,sinϕ̂=icosϕ̂ρ̂.
(A18)

Surprisingly, because the projections of the momentum operator onto non-Cartesian unit-vector operators are not canonical momenta, we need to compute their commutators, which do not vanish, in general. We find that

p̂r,p̂θ=ir̂p̂θ,
(A19)
p̂r,p̂ϕ=ir̂p̂ϕ,
(A20)

and

p̂θ,p̂ϕ=icotθ̂r̂p̂ϕ.
(A21)

In two dimensions, there are only a few changes. We define êρ=(r̂xex+r̂yey)/ρ̂ and êϕ=(r̂yex+r̂xey)/ρ̂, the latter vector being the same as in three dimensions. Using the trigonometric operators, these become

p̂ρ=cosϕ̂p̂x+sinϕ̂p̂yi2ρ̂
(A22)

and

p̂ϕ=sinϕ̂p̂x+cosϕ̂p̂y
(A23)

after evaluating the quantum correction for the radial momentum operator. The commutators are similar, and we report them here: first, for the radial momentum, which satisfies

p̂ρ,ρ̂=i,
(A24)

while p̂ρ commutes with cosϕ̂ and sinϕ̂, and next, for the ϕ-component of the momentum

p̂ϕ,cosϕ̂=isinϕ̂ρ̂andp̂ϕ,sinϕ̂=icosϕ̂ρ̂.
(A25)

We also have p̂ϕ,ρ̂=0. Finally, we compute the commutation relation between the components of the momentum operator. This yields

p̂ρ,p̂ϕ=iρ̂p̂ϕ.
(A26)

The last set of identities we derive in this appendix is the conversion of the kinetic-energy operator into its radial and angular components. Beginning in three dimensions, we find that

p̂r2=1r̂r̂xp̂x+r̂yp̂y+r̂zp̂zi1r̂r̂xp̂x+r̂yp̂y+r̂zp̂zi=1r̂2r̂x2p̂x2+r̂y2p̂y2+r̂z2p̂z2+2r̂xr̂yp̂xp̂y+2r̂yr̂zp̂yp̂z+2r̂zr̂xp̂xp̂x2ir̂xp̂x+r̂yp̂y+r̂zp̂z.
(A27)

Next, using the standard definition of orbital angular momentum,

L̂α=βγϵαβγr̂βp̂γ,
(A28)

we find that

L̂L̂r̂2=1r̂2r̂xp̂yr̂yp̂xr̂xp̂yr̂yp̂x+r̂yp̂zr̂zp̂yr̂yp̂zr̂zp̂y+r̂zp̂xr̂xp̂zr̂zp̂xr̂xp̂z=1r̂2r̂y2+r̂z2p̂x2+r̂x2+r̂z2p̂y2+r̂x2+r̂y2p̂z22r̂xr̂yp̂xp̂y2r̂yr̂zp̂yp̂z2r̂zr̂xp̂xp̂x+2ir̂xp̂x+r̂yp̂y+r̂zp̂z.
(A29)

Adding together our results for p̂r2 and L̂L̂/r̂2 yields

12μ(p̂x2+p̂y2+p̂z2)=12μp̂r2+L̂L̂r̂2.
(A30)

Note that there are no quantum corrections in this case.

In two dimensions, we find that

p̂ρ2=1ρ̂(r̂xp̂x+r̂yp̂yi2)1ρ̂(r̂xp̂x+r̂yp̂yi2)=1ρ̂2r̂x2p̂x2+r̂y2p̂y2+2r̂xr̂yp̂xp̂yi(r̂xp̂x+r̂yp̂y)+24
(A31)

and

L̂z2ρ̂2=1ρ̂2r̂y2p̂x2+r̂x2p̂y22r̂xr̂yp̂xp̂y+i(r̂xp̂x+r̂yp̂y).
(A32)

Again, adding these two together yields

12μ(p̂x2+p̂y2)=12μp̂ρ2+L̂z2142ρ̂2.
(A33)

In this case, there is a quantum correction (namely, the term proportional to 2).

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