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3.10 Time dependence of the expectation value 97
È
parameter, so that the time-independent Schrodinger equation (3.58) may be
written as
^
H(ë)ø n (ë) E n (ë)ø n (ë) (3:66)
^
The expectation value of H(ë) is, then
^
E n (ë) hø n (ë)jH(ë)jø n (ë)i (3:67)
where we assume that ø n (ë) is normalized
hø n (ë)jø n (ë)i 1 (3:68)
To obtain the Hellmann±Feynman theorem, we differentiate equation (3.67)
with respect to ë
* +
d d ^
dë E n (ë) ø n (ë) dë H(ë) ø n (ë)
* + * +
d d
^
^
ø n (ë) H(ë) ø n (ë) ø n (ë) H(ë) ø n (ë) (3:69)
dë dë
^
Applying the hermitian property of H(ë) to the third integral on the right-hand
side of equation (3.69) and then applying (3.66) to the second and third terms,
we obtain
* +
d d
^
E n (ë) ø n (ë) H(ë) ø n (ë)
dë dë
" * + * +#
d d
E n (ë) ø n (ë) ø n (ë) ø n (ë) ø n (ë) (3:70)
dë dë
The derivative of equation (3.68) with respect to ë is
* + * +
d d
dë ø n (ë) ø n (ë) ø n (ë) dë ø n (ë) 0
showing that the last term on the right-hand side of (3.70) vanishes. We thereby
obtain the Hellmann±Feynman theorem
* +
d d
^
E n (ë) ø n (ë) H(ë) ø n (ë) (3:71)
dë dë
3.10 Time dependence of the expectation value
The expectation value hAi of the dynamical quantity or observable A is, in
general, a function of the time t. To determine how hAi changes with time, we
take the time derivative of equation (3.46)
