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Stationary States
An eigenstate of the energy operator
corresponding to the eigenvalue satisfies

(293) 
It is evident that this equation can be solved by writing

(294) 
where is a properly normalized stationary (i.e., nontimevarying) wavefunction. The wavefunction
corresponds to a socalled stationary state, since
the probability density
is nontimevarying. 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:

(295) 
This is known as the timeindependent Schrödinger equation.
More generally, this equation takes the form

(296) 
where is assumed not to be an explicit function of .
Of course, the satisfy the usual orthonormality condition:

(297) 
Moreover, we can express a general wavefunction as a linear combination
of energy eigenstates:

(298) 
where

(299) 
Here, is the probability that a measurement of the energy will
yield the eigenvalue . 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.
Subsections
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Richard Fitzpatrick
20100720