Physics 225/315
March 5 2008
One Dimensional QM
Recall
T
=
p
2
/
2
m
in classical physics, The de Broglie relation is
p
=
h/λ
= ¯
hk
.
p
=
√
2
mE
and
k
= 2
π/λ
=
q
2
mE/
¯
h
2
The momentum operator in the
x
representation is
¯
h
i
d
dx
. The kinetic energy part of the Hamiltonian is
T
=
p
2
/
2
m
=

¯
h
2
2
m
d
2
dx
2
In general
H
=
T
+
V
(
x
) where the potential energy depends on
x
but we assume is inde
pendent of time.
Notice that the momentum operator,
p
, does not commute with
x
; that is
xpψ
(
x
) is not the
same as
pxψ
(
x
) since
p
involves a derivative. In fact the commutator is [
x, p
] =
i
¯
h
. As in
the previous case of the spin operators this failure to commute is related to the uncertainty
principle relating
x
and
p
.
The Schrodinger equation
H
Ψ(
x, t
) =
i
¯
h
∂
Ψ(
x, t
)
∂t
becomes
ˆ

¯
h
2
2
m
∂
2
∂x
2
+
V
(
x
)
!
Ψ(
x, t
) =
i
¯
h
∂
Ψ(
x, t
)
∂t
Separate the variables so Ψ(
x, t
) =
ψ
(
x
)
φ
(
t
).
Then the time equation has the solution
φ
(
t
) =
e

iEt/
¯
h
where
E
, the separation constant, is the eigenvalue of the Hamiltonian, the energy. The
x
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 ROTHBERG
 Energy, Kinetic Energy, Momentum, Uncertainty Principle, 2m, wave function, 2 2m

Click to edit the document details