Complex numbers are broadly used in physics, normally as a calculation tool that makes things easier due to Euler's formula. In the end, it is only the real component that has physical meaning or the two parts (real and imaginary) are treated separately as real quantities. However, the situation seems to be different in quantum mechanics, since the imaginary unit i appears explicitly in its fundamental equations. From a learning perspective, this can create some challenges to newcomers. In this article, four conceptually different justifications for the use/need of complex numbers in quantum mechanics are presented and some pedagogical implications are discussed.

1.
Salomon
Bochner
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The significance of some basic mathematical conceptions for physics
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2.

This struggle is well documented in Schrödinger's letters and papers. Here, we give two examples. In a letter to Lorentz on June 6, 1926, Schrödinger wrote: “What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. ψ is surely fundamentally a real function.” The last paragraph of his fourth communication to the Annalen der Physik also illustrates this feeling of puzzlement and frustration: “Meantime, there is no doubt a certain crudeness in the use of a complex wave function.”

3.
C. N.
Yang
, “
Square root of minus one, complex phases and Erwin Schrödinger
,” in
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, edited by
C. W.
Kilmister
(
Cambridge U. P
.,
Cambridge
,
1987
), pp.
53
64
.
4.

In fact, Dirac is quite explicit about that in the fourth edition of his celebrated book The Principles of Quantum Mechanics (p. 20, our emphasis): “Our bra and ket vectors are complex quantities, since they can be multiplied by complex numbers and are then of the same nature as before, but they are complex quantities of a special kind which cannot be split up into real and pure imaginary parts. The usual method of getting the real part of a complex quantity, by taking half the sum of the quantity itself and its conjugate, cannot be applied since a bra and a ket vector are of different natures and cannot be added.”

5.
It is worth mentioning that we are scratching the surface of a much deeper question that has been addressed by many serious researchers on the mathematical foundations of quantum mechanics (e.g., Ref. 6). Among them, the work of Stueckelberg (Ref. 7) stands out as a plausible formulation of quantum mechanics in real Hilbert space (see Ref. 8 for a pedagogical introduction). The interested reader will also find numerous discussions about the topic in internet forums, such as <https://physics.stackexchange.com/questions/32422/qm-without-complex-numbers>.
6.
Felix M.
Lev
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Why is quantum physics based on complex numbers?
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Ernst C. G.
Stueckelberg
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8.
Jan
Myrheim
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Quantum mechanics on a real Hilbert space
,” e-print arXiv:quant-ph/9905037.
9.
A broad literature review shows very few attempts to explain the need for complex numbers in quantum mechanics, both in introductory level textbooks or physics education journals. An exception is the paper “Why i?” [
W. E.
Baylis
et al.,
Am. J. Phys.
60
(
9
),
788
797
(
1992
)], which actually deals with the question of why complex numbers are useful for physics in general. This paper is based on the geometric algebra by David Hestenes, who proposes a new mathematical formalism for physics [Am. J. Phys. 71(2), 104 (2003)]. Because of its geometrical interpretation, the reasons for “Why i?” in quantum mechanics enabled by this formalism are very convincing and it is definitely worthwhile pursuing this project. However, here we are concerned with physics instructors who use more standard approaches.
10.
Ramamurti
Shankar
,
Fundamentals of Physics II: Electromagnetism, Optics, and Quantum Mechanics
(
Yale U.P
.,
New Haven
,
2016
).
11.

If one takes the precise definition of uncertainty in quantum theory, the cosine function does not violate the uncertainty principle, since its uncertainty is infinite (plane wave). Nevertheless, the general argument that this function does provide information about position still holds. I am indebted to two reviewers for pointing this out.

12.
David
Bohm
,
Quantum Theory
(
Prentice-Hall
,
New York
,
1951
).
13.

The main argument would remain valid in the general case.

14.

From a pedagogical perspective, it would be important to show where this equation comes from. Schrödinger derived it from Hamilton's optical-mechanical analogy (see Ref. 15). For our purpose here, it is worth stressing that one does not need to use the complex exponential in this derivation, i.e., the same expression is obtained if one assumes $ψ(x,t)=Ψ(x)· cos (E/ℏt)$.

15.
Erwin
Schrödinger
,
Four Lectures on Wave Mechanics
, delivered at the Royal institution, London, on 5th, 7th, 12th, and 14th March, 1928 (
Blackie & son limited
,
London, Glasgow
,
1928
).
16.

If we had represented ψ as a purely real periodic function, e.g., $ψ(x,t)=Ψ(x)· cos (E/ℏ)t$, then we would need to derive it twice with respect to time to isolate the energy parameter ($∂2ψ/∂t2=−(E2/ℏ2)ψ$). In order to substitute this in Eq. (7), we would also need to square the whole equation and would end up with a rather complicated fourth order equation that looks like $((∂2/∂x2)−(2m/ℏ)V)2ψ+(4m2/ℏ2)(∂2ψ/∂t2)=0$. Curiously, this fourth-order equation was the one Schrödinger initially claimed to be “the uniform and general wave equation for the scalar field ψ,” since apparently it would be possible to consider ψ as a real function. One can hypothesize that Schrödinger did this in order to avoid an explicit i in his fundamental equation. Dealing with this matter is beyond the scope of this article, but the interested reader can find deeper discussions about the implications of a real wave function in the continuation of Bohm's argument (Ref. 12) and in Ref. 17.

17.
Robert L. W.
Chen
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Derivation of the real form of Schrödinger's equation for a nonconservative system and the unique relation between Re(ψ) and Im(ψ)
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Jun John
Sakurai
,
(
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Boston
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1967
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19.
D. H.
McIntyre
,
C. A.
Manogue
, and
J.
Tate
,
(
,
Boston
,
2012
).
20.
John S.
Townsend
,
A Modern Approach to Quantum Mechanics
, 2nd ed. (
University Science Books
,
Mill Valley, CA
,
2013
).
21.

We will assume that a and c are real because we want to see if we can get away without complex numbers. A more general treatment of the situation is presented in Ref. 20.

22.

The symbol $≐$ is used to indicate “is represented by.” One cannot set the ket equal to the column vector, because the former is an abstract vector and the latter is its representation in a given basis.

23.
Paul J.
Nahin
,
Imaginary Tale: The Story of
$−1$ (
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.,
Princeton
,
1998
).
24.
Lucien
Hardy
, “
Quantum physics from five reasonable axioms
,” e-print arXiv:quant-ph/0101012v4.
25.
Artur
Ekert
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Complex and unpredictable Cardano
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47
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2008
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26.

There is a great deal of gymnastics to calculate the square root of a square matrix. For this particular case, the fact that the determinant of T1 is negative already guarantees that the components of its square roots are complex numbers with non-zero imaginary parts (see Ref. 27).

27.
Bernard W.
Levinger
, “
The square root of a 2 × 2 matrix
,”
Math. Mag.
53
(
4
),
222
224
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1980
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28.

See, for instance, Wigner's unreasonable effectiveness paper [Pure Appl. Math. 13, 1–14 (1960)] and the debates related to it.