ψ(r,t) =ψ(r)e−iωt=ψ(r)e−iEt/Z as ω=E/Z (9.34)
With this substitution we can write:
(9.35)
Using the relationship that:
(9.36)
gives:
(9.37)
(9.38)
(9.39)
This last equation is the time-independent form of Schrödinger’s equation
and will be the form that we use to solve various applications including
the hydrogen atom (Chapter 11).
The two relationships in quantum mechanics, the de Broglie and
Schrödinger equations, are consistent with each other, as can be seen by
substituting into Schrödinger’s equation a typical wave. For simplicity, one-
dimensional expressions can be used. The oscillation behavior of a wave
can be expressed by a number of equivalent formulations, for example:
(9.40)
The difference for these two expressions is the phase; that is, whether
the wave has a peak at 0 or 90°. Eqn 9.40 can be rewritten in terms of
exponentials since:
eikx=cos(kx) +isin(kx)
and
e−ikx=cos(kx) −isin(kx) (9.41)
ψ()x =Asin( )kx or Acos( )kx where k=
2 π
λEr
m
ψψψ()=− ∇()rVrr+ () ()Z^22
2
ei
iE
re
m−−iEt ⎡ − iEt
⎣⎢
⎢
⎤
⎦
⎥
⎥
//=−∇
()
ZZZ ()
Z
Z
ψ2
2⎡ (^22) ψψ()rVrr+ () ()
⎣
⎢
⎢
⎤
⎦
⎥
⎥
ψ
∂
∂
()ri ( //) ψ()
tee
mZ iEt iEt rV−−ZZZ
=−∇+
2
2
2(() ()rrψ⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
d
dxeaeax= axi
t
re
mZ iEt reiEt∂ ZZZ
∂
[()ψψ−−//]=− ∇[() ]+2
2
2Vr()[ ()ψre−iEt/Z]