We give a simple argument to show that the nth wavefunction for the one-dimensional Schrödinger equation has n1 nodes. We also show that if n1<n2, then between two consecutive zeros of ψn1, there is a zero of ψn2.

1.
R. P.
Feynman
,
Statistical Mechanics: A Set of Lectures
(
Addison-Wesley
, Reading, MA,
1998
).
See also Appendix A in the Ph.D. thesis by
M.
Cohen
, “
The energy spectrum of the excitations in liquid helium
,”
http://etd.caltech.edu/etd/available/etd-03192004-153651/.
2.
J.
Mur-Petit
,
A.
Polls
, and
F.
Mazzanti
, “
The variational principle and simple properties of the ground-state wave function
,”
Am. J. Phys.
70
(
8
),
808
810
(
2002
).
3.
L. D.
Landau
and
E. M.
Lifschitz
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Quantum Mechanics: Non-Relativistic Theory
(
Butterworth-Heinemann
, Oxford,
1981
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4.
A.
Galindo
and
P.
Pascual
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Quantum Mechanics
(
Springer
, Berlin,
1991
), Vol.
1
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5.
R.
Courant
and
D.
Hilbert
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Methods of Mathematical Physics
(
Wiley-Interscience
, New York,
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6.
P.
Hartman
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Ordinary Differential Equations
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Wiley
, New York,
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7.
Géza
Makay
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A simple proof for Sturm’s separation theorem
,”
Am. Math. Monthly
99
,
218
219
(
1992
).
8.

We will call the ground state function ψ1 instead of the more common notation ψ0.

9.

The prime denotes the spatial derivative.

10.
S.
Flügge
,
Practical Quantum Mechanics
(
Springer
, New York,
1998
), Problems 25 and 26.
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