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Material Type: Notes; Professor: Bromley; Class: QUANTUM MECHANICS; Subject: Physics; University: San Diego State University; Term: Spring 2006;
Typology: Study notes
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Schr¨
odinger Eqn. Soln [end of Section 4.3]
Quantum State time evolution
Free Particle [Section 5.1]
(if time) Wavepacket propagation [Section 5.1]
i ℏ
|
dtdψ
( t ) 〉 = ˆ
H | ψ ( t ) 〉 ˆ
when
is Hermitian eigensolutions must exist!
| ψ ( t ) 〉 = ∑ | E
ψ
e
−
iEt/
ℏ
a
E
(^) (0)
e
−
iEt/
ℏ | E
Eg. for practice, spin-
2 1
particle in magnetic field:
γB
0 S
z
γB
0 ℏ
Eigenstates (
z +
z −
) are the same as those of
z
with eigenvalues
z +
2 1
(^) γB
0 ℏ
and
z −
2 1
(^) γB
0 ℏ
.
If initial state is eigenstate of Hamiltonian
ψ
then we calculate the state evolution simply as
| ψ ( t ) 〉 = ∑ | E
ψ
e
−
iEt/
ℏ
e
−
iEt/
ℏ | E 〉 = | E ( t ) 〉
BUT if we do any other measurement
of this state:
ω
i , t
ω i | ψ ( t )
2
ω i | E ( t )
2
e
−
iEt/
ℏ 〈 ω i | E
2
ω
i | E
2
ω
i , t
ie the probability of measuring
ω
i
is constant
If initial state is eigenstate of
z
ψ
z +
but we place the magnetic field along y direction:
γB
0 S
y
γB
0 ℏ
i
i
So the state evolution via expansion in eigenvectors of
| ψ ( t ) 〉 = c | E
y +
d
| E
y −
c
i^1
d
i
After some time we then do measurements of
z
Schr¨
odinger equation for free particle
i ℏ
d
dt
ψ ( t ) 〉 = ˆ
H | ψ ( t ) 〉 =
ˆp 2
2 m
ψ
t ) 〉
We already know eigensolns / stationary modes, ie | ψ ( t ) 〉 = | E 〉 e −
iEt/
ℏ
ˆp 2
2 m | E 〉 = E | E 〉
noting that an eigenstate
ˆp | p 〉 = p | p 〉
is also an
eigenstate
ˆp 2 | p 〉 = p
ˆp | p 〉 = p 2 | p 〉
we can trial:
ˆp 2
2 m | p 〉 = E | p 〉 ⇒ ( p 2
2 m − E ) | p 〉
⇒ p = ± √ 2
mE
each eigenvalue
has two orthogonal eigenstates! so
α | E + 〉 + β | E − 〉 = α | p + 〉 + β | p − 〉 = | E 〉
ie. probability particle measured moving left or right
Note that that proof didn’t rely on a particular space!
but for now choose position space (
ˆp
i ℏ
d
dx
ˆp 2
2 m ψ = − ℏ 2
m
d
2 ψ
dx
2
Eψ
thus
d
2 ψ
dx
2 = − k 2 ψ
we’ve already found solns, to which add time bits
ψ
x
) =
Ae
ikx
Be
−
ikx
ψ
x, t
Ae
ikx
Be
−
ikx
e
−
iEt/
ℏ
ψ
x, t
Ae
ik
( x −
ℏ k
2 m
(^) t )
Be
−
ik
( x
ℏ k
2 m
(^) t )
Waves have momentum
p
k
(cred to de Broglie).
so-called
plane-waves
have wavenumber
k
mE/