4.12: Stationary States
( \newcommand{\kernel}{\mathrm{null}\,}\)
An eigenstate of the energy operator H≡iℏ∂/∂t corresponding to the eigenvalue Ei satisfies
iℏ∂ψE(x,t,Ei)∂t=EiψE(x,t,Ei)
It is evident that this equation can be solved by writing
ψE(x,t,Ei)=ψi(x)e−iEit/ℏ
where ψi(x) is a properly normalized stationary (i.e., non-time-varying) wavefunction. The wavefunction ψE(x,t,Ei) corresponds to a so-called stationary state, since the probability density |ψE|2 is non-time-varying. Note that a stationary state is associated with a unique value for the energy. Substitution of the above expression into Schrödinger's equation (137) yields the equation satisfied by the stationary wavefunction:
ℏ22md2ψidx2=[V(x)−Ei]ψi
This is known as the time-independent Schrödinger equation. More generally, this equation takes the form
Hψi=Eiψi
where is assumed not to be an explicit function of
. Of course, the ψi satisfy the usual orthonormality condition:
∫∞−∞ψ∗iψjdx=δij
Moreover, we can express a general wavefunction as a linear combination of energy eigenstates:
ψ(x,t)=∑iciψi(x)e−iEit/ℏ
where
ci=∫∞−∞ψ∗i(x)ψ(x,0)dx
Here, |ci|2 is the probability that a measurement of the energy will yield the eigenvalue Ei. Furthermore, immediately after such a measurement, the system is left in the corresponding energy eigenstate. The generalization of the above results to the case where has continuous eigenvalues is straightforward.
If a dynamical variable is represented by some Hermitian operator which commutes with
(so that it has simultaneous eigenstates with
), and contains no specific time dependence, then it is evident from Eqs. (297) and (298) that the expectation value and variance of
are time independent. In this sense, the dynamical variable in question is a constant of the motion.
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)