This wave function fulfills
HðR; t; t
0
Þ À i h
@
@t
! u g ðR; t
0
Þe
ixðt
0 Þt
u u ðR; t
0
Þ
!
¼ E F ðt
0
Þ
u g ðR; t
0
Þe
ixðt
0 Þt
u u ðR; t
0
Þ
!
ð16Þ
or also
HðR; t; t
0
Þ
u g ðR; t
0
Þe
ixðt
0 Þt
u u ðR; t
0
Þ
!
¼ i h
ðixðt
0
Þu g ðR; t
0
Þe
ixðt
0 Þt
0
!
þ E F ðt
0
Þ
u g ðR; t
0
Þe
ixðt
0 Þt
u u ðR; t
0
Þ
!
ð17Þ
Because in the Hamiltonian there is no derivative with respect to either t ot t
0 we
can let t
0 to be equal to t, so that
HðR; tÞ
u g ðR; tÞe
ixðtÞt
u u ðR; tÞ
!
¼ i h
ixðtÞu g ðR; tÞe
ixðtÞt
0
!
þ E F ðtÞ
u g ðR; tÞe
ixðtÞt
u u ðR; tÞ
!
ð18Þ
We look now for a solution with the form
UðR; tÞ ¼ AðtÞe
À
i
h
R t
0
E F ðt
0 Þdt
0
u g ðR; tÞe
ixðtÞt
u u ðR; tÞ
!
ð19Þ
In order to obtain an equation for the amplitude AðtÞ, we apply to HðR; tÞUðR; tÞ
the row vector
u g ðR; tÞe
ÀixðtÞt
; u u ðR; tÞ
h
i
ð20Þ
In the definition of this “bra” there is no complex conjugation of the coordinate
dependent functions (cf the c-product of reference [22] valid for biorthogonal
functions), but complex conjugation of the time dependent functions (cf the
F-product of reference [23]). We adopt for the scalar product the notation j
ð Þ. There
is obtained
u g ðR; tÞe
ÀixðtÞt
; u u ðR; tÞ
h
i
jHðR; tÞUðR; tÞ
¼ AðtÞe
À
i
h
R t
0
E F ðt
0 Þdt
0
½i hixðtÞðu g ju g Þ þ E F ðtÞ½ðu g ju g Þ þ ðu u ju u ފ
Â
Ã
ð21Þ
where, for instance, ðu g ju g Þ stands for
142
R. Lefebvre