Download Quantum Mechanics Lecture 5 Dr. Mauro Ferreira

Quantum Mechanics
Lecture 5
Dr. Mauro Ferreira
E-mail: [email protected]
Room 2.49, Lloyd Institute
Time evolution of expectation values
Consider an operator Â. The expectation value <A> is given by
!A" =
!
∞
dx Ψ∗ (x, t) Â Ψ(x, t)
taking the derivative...
−∞
d
!A" =
dt
!
∞
∂Ψ∗ (x, t)
dx
 Ψ(x, t) +
∂t
−∞
! ∞
∂ Â
∗
Ψ(x, t) +
dx Ψ (x, t)
∂t
−∞
!
∞
∂Ψ(x, t)
dx Ψ (x, t) Â
∂t
−∞
∗
∂
Bearing in mind that i! Ψ(x, t) = ĤΨ(x, t)
∂t
d
1
∂ Â
!A" =
![Â, Ĥ]" + !
"
dt
i!
∂t
If  is not explicitly dependent on time
1
d
!A" =
![Â, Ĥ]"
dt
i!
The commutator [A,H] determines the time evolution of the
expected value of the quantity <A>
If  commutes with the Hamiltonian, the quantity <A> is constant
The knowledge of all eigenstates of a conservative Hamiltonian allows us
to fully determine how the system evolves in time.
! r) ψ(!r) = E ψ(!r)
H(!
(Time-independent Schroedinger Eq.)
Eigenvalues and eigenfunctions of conservative Hamiltonians are
key quantities in QM and will be often used here
e
m
o
h
e
k
a
T
e
g
a
s
s
e
m
In summary, we have seen that
• Wave functions (WF) are used to describe the state of a
system;
• Probabilistic nature of QM is in extracting info from the
WF;
• Physical quantities are represented by linear operators.
Their eigenvalues provide the allowed values for those
quantities;
• Measurement sensitivity is reflected in the action of those
operators. In particular, the commutator of two different
operators define whether or not the corresponding
quantities can be simultaneously known;
• Time evolution is fully deterministic and described by the
Schroedinger Eq.
Example:
Use the property
d
!p"
!x" =
dt
m
d
1
∂ Â
!A" =
![Â, Ĥ]" + !
"
dt
i!
∂t
to show that
1
d
1
p̂2
!x" =
![x̂, Ĥ]" =
![x̂,
+ V̂ (x)]"
dt
i!
i!
2m
0
1
p̂2
1
=
![x̂,
]" +
![x̂, V̂ (x)]"
i!
2m
i!
1
=
{![x̂, p̂] p̂" + !p̂ [x̂, p̂]" }
2i!m
bearing in mind that
[x̂, p̂] = i!
d
!p"
!x" =
dt
m
Sim
ilar
to c
mec
lass
ical
han
ics
Example:
d
Evaluate !p"
dt
d
1
∂ p̂
!p" = ![p̂, Ĥ]" + ! "
dt
i!
∂t
0
Wh
at’s
with the re
l
atio
c
l
a
mec ssic
n
han al
ics?
0
! d
p̂2
! dV̂ (x)
[p̂, Ĥ] = [p̂,
] + [p̂, V̂ (x)] = [ , V̂ (x)] =
i dx
2m
i dx
d
dV (x)
!p" = !−
"
dt
dx
Ehrenfest’s theorem states
that expectation values
obey classical laws
Time-independent Schroedinger Eq.
!2 2
−
∇ ψ("r) + V ("r)ψ("r) = E ψ("r)
2m
(3-dimensional Eq.)
!2 d2 ψ(x)
−
+ V (x)ψ(x) = E ψ(x) (1-dimensional Eq.)
2
2m dx
a simple manipulation of the Schroedinger Eq. gives
d2 ψ(x)
2m
= 2 (V (x) − E)ψ(x)
2
dx
!
±αx
ψ(x) ≈ e
ψ(x) ≈ e±iqx
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