Quantum Mechanics: Uncertainty Principle and Non-commuting Operators - Prof. M. W. Bromley, Study notes of Quantum Mechanics

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Lecture 9 Outline - observables / S.E.

QM postulates [Section 4.1 and 4.2]:

Incompatible observables / Uncertainty principle

eg. Gaussians and Stern-Gerlach

Schr¨

odinger Eqn. [Section 4.3]

Generalised Uncertainty - eg. Gaussian

σ

Ω 2 σ

Λ 2

(^21) i 〈 [Ω

Λ]

non-zero for non-commuting operaters eg.

[ˆ

x,

ˆp ] =

i ℏ ,

Last time: Gaussian

〈 x | ψ 〉 = ψ ( x

πσ

2 ) −

4 1 e^ − ( x − a ) 2 / 2 σ 2

〈 x 〉 = 〈 ψ |

ˆx | ψ 〉 = ∫ ∞

−∞

ψ

| x

〉〈

x

| ˆx | ψ

〉 dx

∞

−∞

ψ

∗ ( x

) xψ

x

) dx

a

Recall

K

-defn of eigenfunctions

ψ

k ( x

) =

〈 x | k 〉 =

Ae

ikx

φ

( p ) =

〈 p | ψ 〉 = ∫ ∞

−∞

p | x 〉〈

x | ψ

〉 dx

∞

−∞

ψ

p ∗ (^) ( x ) ψ ( x )

dx

∞

−∞

e −

ipx/

ℏ

√ 2 π ℏ e − ( x − a ) 2 / 2 σ 2

πσ

2 ) 41

dx

σ

2

π

ℏ 2 )

41

e −

ipa/

ℏ e − p 2 σ 2 / 2 ℏ 2

So...

Non-commuting operators - Discrete

σ

Ω 2 σ

Λ 2

(^21) i 〈 [Ω

Λ]

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...

(lack of ) Simultaneous Eigenfunctions

all are Hermitian with eigenvalues

and

Diagonalisation

ie. all are Hermitian (measure spin in

x, y, z

dirs)

choice of phase...

ie. Ivan’s innocent question

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.

Schr¨

odinger Eqn - General soln

i ℏ | dtdψ

( t ) 〉 = ˆ

H | ψ ( t ) 〉 ˆ

H | E 〉 = E | E 〉

when

H

is Hermitian eigensolutions must exist!

H

eigenproblem is

time-independent

Schr¨

odinger Eqn.

| ψ ( t ) 〉 = ∑

|^

E

E | ψ ( t ) 〉 = ∑

a^ E (^) ( t ) | E

Now rewrite

i ℏ

dtd

(^) | ψ

( t ) 〉 −

H | ψ ( t ) 〉 = | 0 〉

| 0 〉 = ( i ℏ

dt d

H ) | ψ ( t ) 〉 = ∑ ( i ℏ

da

E (^) ( t )

dt

Ea

E (^) ( t ) ) | E n 〉

By linear independence defn (

| 0 〉 = ∑ i c i | i 〉

iff

a i

= 0

da

E (^) ( t )

dt

iE ℏ

a E (^) ( t )

solns

a E (^) ( t ) =

a E (^) (0)

e − iEt/

ℏ

| ψ ( t ) 〉 = ∑

a^

E

(^) (0)

e − iEt/

ℏ | E 〉 = ∑ | E

E

ψ

(0)

e −

iEt/

ℏ