3.10: Stationary States
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Apr 1, 2025
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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, because 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 previous expression into Schrödinger’s equation ([e3.1]) 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 H is assumed not to be an explicit function of t. 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 previous results to the case where H has continuous eigenvalues is straightforward.
If a dynamical variable is represented by some Hermitian operator A that commutes with H (so that it has simultaneous eigenstates with H), and contains no specific time dependence, then it is evident from Equations ([e4.157]) and ([e4.158]) that the expectation value and variance of A are time independent. In this sense, the dynamical variable in question is a constant of the motion.
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