V (x)= − e2
4π 0|x|. (1.64)
Schrửdinger’s solution of the hydrogen atom will be discussed in Chap.7.
1.7 Interpretation of Schrửdinger’s Wave Function
The Schrửdinger equation was a spectacular success right from the start, but it was not immediately clear what the physical meaning of the complex wave function ψ(x, t )is. A natural first guess would be to assume that|ψ(x, t )|2corresponds to a physical density of the particle described by the wave functionψ(x, t ). In this interpretation, an electron in a quantum state ψ(x, t )would have a spatial mass densitym|ψ(x, t )|2and a charge density−e|ψ(x, t )|2. This interpretation would imply that waves would have prevailed over particles in wave-particle duality.
However, quantum jumps are difficult to reconcile with a physical density interpretation for|ψ(x, t )|2, and Born, Bohr and Heisenberg developed a statistical interpretation of the wave function which is still the leading paradigm for quantum mechanics. Already in June 1926, the view began to emerge that the wave function ψ(x, t )should be interpreted as aprobability density amplitude2in the sense that
2Schrửdinger [151, paragraph on pp. 134–135, sentences 2–4]: “ψψis a kind ofweight function in the configuration space of the system. The wave mechanical configuration of the system is a superposition of many, strictly speaking ofall, kinematically possible point mechanical configurations. Thereby each point mechanical configuration contributes with a certainweight to the true wave mechanical configuration, where the weight is just given byψψ.” Of course, a weakness of this early hint at the probability interpretation is the vague reference to a “true wave mechanical configuration”. A clearer formulation of this point was offered by Born, see the reference to Born’s work below. While there may have been early agreement on the importance of a probabilistic interpretation, the question of the concept which underlies those probabilities was a contentious point between Schrửdinger, who at that time may have preferred to advance a de Broglie type pilot wave interpretation, and Bohr and Born and their particle-wave complementarity interpretation. Indeed, Schrửdinger himself was intrigued by the possibility of the wave function describing continuous electronic oscillations in atoms without quantum jumps, see pp. 121 and 129–130 inloc. cit. and Schrửdinger’s papers in the British Journal for the Philosophy of Science 3, 109 (1952);ibid.233 (1952)). This as well as concerns about probabilistic interpretations of superpositions of states ultimately made him sceptic regarding the probabilistic interpretation of wave functions and the concept of elementary particles. Nevertheless, in the end the probabilistic complementarity picture prevailed: There are fundamental degrees of freedom with certain quantum numbers. These degrees of freedom are quantal excitations of the vacuum, and mathematically they are described by quantum fields. Depending on the way they are probed,
PV(t )=
V
d3x |ψ(x, t )|2 (1.65) is the probability to find a particle (or rather, an excitation of the vacuum with minimal energymc2and certain other quantum numbers) in the volumeV at time t. This equation implies that|ψ(x, t )|2is theprobability densityto find the particle in the locationxat timet. The expectation value for the location of the particle at timetis then
x(t )=
d3x x|ψ(x, t )|2, (1.66) where integrals without explicit limits are taken over the full range of the integration variable, i.e. here over all ofR3. Many individual particle measurements will yield the locationxwith a frequency proportionally to|ψ(x, t )|2, and averaging over the observations will yield the expectation value (1.66) with a variance e.g. for thex coordinate
x2(t )= (x− x)2(t )= x2(t )− x2(t )
=
d3xx2|ψ(x, t )|2−
d3xx|ψ(x, t )|2 2
. (1.67)
This interpretation of the relation between the wave function and particle properties was essentially proposed by Max Born in the paper where he invented quantum mechanical scattering theory [17].
The Schrửdinger equation (1.2) implies a local conservation law for probability
∂
∂t |ψ(x, t )|2+∇ãj(x, t )=0 (1.68) with the probability current density
j(x, t )= ¯h 2im
ψ+(x, t )ã∇ψ(x, t )−∇ψ+(x, t )ãψ(x, t )
. (1.69)
The conservation law (1.68) is important for consistency of the probability interpretation of Schrửdinger theory. We assume that the integral
P (t )=
d3x |ψ(x, t )|2 (1.70)
they exhibit wavelike or corpuscular properties, and quantum states represent probability densities for the observation of physical properties of these excitations. Whether or not to denote these excitations as particles is a matter of convenience and tradition.
1.7 Interpretation of Schrửdinger’s Wave Function 21
over R3 converges. A priori this should yield a time-dependent function P (t ).
However, Eq. (1.68) implies
d
dtP (t )=0, (1.71)
whenceP (t ) ≡ P is a positive constant. This allows for rescaling ψ(x, t ) → ψ(x, t )/√
P such that the new wave function still satisfies equation (1.2) and yields a normalized integral
d3x |ψ(x, t )|2=1. (1.72) This means that the probability to find the particle anywhere at timet is 1, as it should be. Equations (1.65) and (1.66) make sense only in conjunction with the normalization condition (1.72)
We can also substitute the Schrửdinger equation or the local conservation law (1.68) into
p(t )=md
dtx(t )=m
d3x x∂
∂t |ψ(x, t )|2 (1.73) to find
p(t )=
d3xψ+(x, t )h¯
i∇ψ(x, t ). (1.74) Equations (1.66) and (1.74) tell us how to extract particle like properties from the wave functionψ(x, t ). At first sight, Eq. (1.74) does not seem to make a lot of intuitive sense. Why should the momentum of a particle be related to the gradient of its wave function? However, recall the Compton-de Broglie relationp = h/λ.
Wave packets which are composed of shorter wavelength components oscillate more rapidly as a function ofx, and therefore have a larger average gradient. Equation (1.74) is therefore in agreement with a basic relation of wave-particle duality.
A related argument in favor of Eq. (1.74) arises from substitution of the Fourier transforms3
ψ(x, t )= 1
√2π3
d3kexp(ikãx)ψ(k, t ), (1.75)
ψ+(x, t )= 1
√2π3
d3k exp(−ikãx)ψ+(k, t ) (1.76)
3Fourier transformation is reviewed in Sect.2.1.
in Eqs. (1.72) and (1.74). This yields
d3k|ψ(k, t )|2=1 (1.77) and
p(t )=
d3khk¯ |ψ(k, t )|2, (1.78) in perfect agreement with the Compton-de Broglie relationp = ¯hk. Apparently
|ψ(k, t )|2is a probability density inkspace in the sense that PV˜(t )=
˜ V
d3k |ψ(k, t )|2 (1.79) is the probability to find the particle with a wave vectorkcontained in a volumeV˜ inkspace.
We can also identify an expression for the energy of a particle which is described by a wave functionψ(x, t ). The Schrửdinger equation (1.2) implies the conservation law
d dt
d3xψ+(x, t )
− ¯h2
2m+V (x)
ψ(x, t )=0. (1.80)
Here it plays a role that we assumed time-independent potential.4 In classical mechanics, the conservation law which appears for motion in a time-independent potential is energy conservation. Therefore, we expect that the expectation value for energy is given by
E =
d3xψ+(x, t )
− ¯h2
2m+V (x)
ψ(x, t ). (1.81) We will also rederive this at a more advanced level in Chap.17. From the classical relation (1.58) between energy and momentum of a particle, we should also have
E = p2
2m + V (x). (1.82)
Comparison of Eqs. (1.74) and (1.81) yields
4Examples of the Schrửdinger equation with time-dependent potentials will be discussed in Chap.13and following chapters.