Quantum Mechanics: Time Evolution, Uncertainty, Schrödinger Equation, and Stern-Gerlach - , Study notes of Quantum Mechanics

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Lecture 9 Outline - Time Evolution

Generalised Uncertainty principle [4.2]

Schr¨

odinger Eqn. [Section 4.3]

Focus on Stern-Gerlach

• Homework

Gaussian in position/momentum space

Gaussian plots

| ψ ( x ) | 2 =

x

| ψ

2

and

| φ ( p ) | 2 =

p

| ψ

2

Just a Fourier transform between them

Non-commuting operators - Discrete

σ

Ω 2

σ

Λ 2

(^21) i

〈 [Ω

Λ]

2

for matrices, if

[Ω

Λ]

cannot find a set of common

eigenvectors that simultaneously diagonalises both...

The S-G

S

x

, S

y , S

z

measurements are the archetype:

S

x

S

y

i

i

S

z

ie. all are Hermitian (measure spin in

x, y, z

dirs)

But they do not commute with each other...

Eigenbasis expansion

Unitary Transformation

Given

S

y ±

can diag

S

y

Schr¨

odinger Eqn

The state vector evolves according to Schr¨

odinger Eqn

i ℏ

dtd

ψ ( t ) 〉 = ˆ

H | ψ ( t ) 〉

Hamiltonian of a Simple Harmonic Oscillator potential

H

classical

p

2

m

mω

2 x

2

H

ˆp 2

m

mω

2

ˆx^

2

Can choose whether to use

x

〉

or

p

〉

as basis

H

2

m

d

2

dx

2

mω

2 x

2

or

H

p

2

m

2 mω

2

d

2

dp

2

depends on which is easier to solve with a wavefunction

We (later on) solve the SHO problem with a 3rd basis.

Determinate States

if

| ψ 〉 = | ω i 〉

, ie. eigenfunction of

where

ω i = 〈 Ω 〉.

These are states where every

measurement

ω

i

σ

Ω 2

− ω i ) 2 〉 = 〈 ψ |

− ω i ) 2 | ψ 〉 = 〈

− ω i ) ψ |

− ω i ) ψ 〉

Note:

is Hermitian, so is

ω

i ) .

The only vector that has zero inner product is

− ω i ) ψ 〉 = | 0 〉

so

Ω | ψ 〉 = ω i | ψ 〉

Time-independent Schr¨

odinger eqn:

H | ψ n 〉 = E n | ψ n 〉

energy of stationary states

ψ

n

|

ˆ

H | ψ n 〉 = E n

is constant

Schr¨

odinger Eqn - Trivial Example

| ψ ( t ) 〉 = ∑ | E n

E

n

| ψ

e

−

iE

n t/

ℏ

a

n

(0)

e

−

iE

n t/

ℏ

| E

n

〉

Eg. for practice, spin-

2 1

particle in magnetic field:

H

γB

0 S

z

γB

0 ℏ

Eigenstates (

E

z +

E

z −

) are the same as those of

S

z

with eigenvalues

E

z +

2 1

(^) γB

0 ℏ

and

E

z −

2 1

(^) γB

0 ℏ

.

Quantum State time-evolution

our spin-

2 1

particle enters magnetic field at

t

and it takes a length of time

t

to pass through.

ψ

a

| E

z +

b | E

z −

a

b

 

b a

 