Spin Dynamics Basic TheoryOperators

Richard GreenSBD Research Group

Department of Chemistry

Objective of this session

Introduce you to operators used in quantum mechanics

Achieve this by looking at:– What is an operator– Linear operators– Hermitian operators– Observables– Angular momentum operators

WARNING – EQUATIONS AHEAD!

What is an operator?

Mathematical entityTransforms one function into another

i.e. operators only act on functions (incl. vectors)

Unit Operator

Og f=

1

means

g f

g f

=

=

Example of operator action

|ψ2 >

|ψ1 >

|f >

Ô

g = Ô|f > = |Ôf >

Operators in quantum mechanics

Distinguish between:

– Not directly observable - wave functions and state vectors;

and

– OBSERVABLES - energy, momentum and other quantities which can be physically measured

Wave functions generally complex

Observables are real numbers – physical measurements

Representation of Observables

In quantum mechanics each observable is represented by a

LINEAR, HERMITIAN OPERATOR

MASSIVELY IMPORTANT

What is a linear operator?

1 2 1 2ˆ ˆ ˆO( ) (O ) (O )f f f fα β α β+ = +

For all functions f1

and f2; and

For all complex constants α and β

Not all operators linear

LINEAR O ( ) ( )d

f x f xdx=

1 2

1 2 1 2( ( ) ( )) ( ( )) ( ( ))df dfd d d

f x f x f x f xdx dx dx dx dxα β α β α β+ = + = +

NON-LINEAR( )O ( ) f x

f x e=

1 1 2 1 2[ ( ) ( )] ( ) ( ) ( ) ( )f x f x f x f x f x f xe e e e eα β α β α β+ = × ≠ +

Hermitian operators definition

ˆ ˆ| O O |f g f g⟨ ⟩=⟨ ⟩

Hermitian operators satisfy this condition

VERY

IMPORTANT

For any normalisable functions f and g

Eigenfunctions, eigenvalues and eigenvalue equations

Of fλ= Eigenvalue equation

f = eigenfunction

λ = eigenvalue (complex constant)

Eigenfunction basis sets

An operator may have more than one eigenfunction and eigenvalue

1 1 1

2 2 2

3 3 3

ˆ

ˆ

ˆ

ˆn n n

Of f

Of f

Of f

Of f

λ

λ

λ

λ

=

=

=

=

⋮

Remember basis sets when we looked at vectors?

They are eigenvectors resulting from solutions of eigenvalue equations.

Hermitian operator properties

Real eigenvalues

Different eigenfunctions (or eigenvectors) corresponding to different eigenvalues are

orthogonal

Operators –in quantum mechanics

Examples– Momentum

– Kinetic Energy

– Potential Energy

ˆ ( )p ix y z

∂ ∂ ∂=− + +

∂ ∂ ∂ℏ

2 2 2 2

2 2 2ˆ ( )

2kinE

m x y z

∂ ∂ ∂=− + +

∂ ∂ ∂

ℏ

ˆ( ) ( )V x V x=

Eigenvalue equation example

Time independent Schröööödingerdingerdingerdinger equationequationequationequation

22

22

[ V( , , )] ( , , )2

eigenvalue of operator [ V( , , )]2

( , , ) corresponding eigenfunction

x y z x y z Em

E x y zm

x y z

ψ ψ

ψ

− ∇ + =

= − ∇ +

=

ℏ

ℏ

Observables (again)

Linear, Hermitian operators allow observable quantities such as energy and spin to be calculated.

The results of observations are the eigenvalues of these operators.

Generally, the eigenvalues of Ô are the only possible outcomes of a measurement of O.

IMPORTANT

Matrix representation of operators

• Square matrices• Compare vectors and functions as column matrices• Vector transformation by operator Ô:

/11 12 1 11

/21 22 2 22

/1 2

n

n

n n nn nn

O O O ff

O O O ff

O O O ff

=

⋯

⋯

⋮ ⋮ ⋮ ⋮⋮

⋯

Orbital angular momentum operators

For information only by way of comparison to what follows

ˆ

ˆ

ˆ

x

y

z

L i y zz y

L i z xx z

L i x yy x

∂ ∂ =− − ∂ ∂ ∂ ∂ =− − ∂ ∂ ∂ ∂ =− − ∂ ∂

ℏ

ℏ

ℏ

Can be derived from classical expressions

Orbital angular momentum commutators

ˆ ˆ ˆ,

ˆ ˆ ˆ,

ˆ ˆ ˆ,

x y z

y z x

z x y

L L i L

L L i L

L L i L

=

=

=

ℏ

ℏ

ℏ

Spin (angular momentum) operators

No classical starting point exists

Approach• Impose experimentally-observed constraints

• Impose need for spin operators to be linear and Hermitian (because a physical property)

• Assume commutation relationships similar to orbital angular momentum obeyed

2±ℏ

Resulting spin operators

• Spin-½ spin states represented by spinors (2 x 1 column matrices)

• Spin operators (observables) represented by 2 x 2 matrices

• 2 x 2 spin operator acting on 2 x 1 spinor produces new 2 x 1 spinor

Resulting general spin operator in spherical coordinates

z

y

x

n

θ

φ

cos sinˆ2 sin cos

i

n i

eS

e

φ

φ

θ θ

θ θ

− = −

ℏ

General representations for arbitrary direction n

cos( / 2) sin( / 2)| and |

sin( / 2) cos( / 2)

i

n ni

e

e

φ

φ

θ θ

θ θ

− ↑ ⟩= ↓ ⟩=

These two vectors provide orthonormal basis for spin space such that any spin state |A> can be written:

1 2| | |n nA c c⟩= ↑ ⟩+ ↓ ⟩

Spin eigenvectors

Pauli spin matrices used in NMR for the x, y and z directions

0 11ˆ90 01 02

01ˆ90 9002

1 01ˆ0 00 12

x

y

z

I

iI

i

I

θ ϕ

θ ϕ

θ ϕ

= = =

− = = =

= = = −

� �

� �

� �

Summary

• Defined operators• Need for operators in quantum mechanics• Observables – linear, Hermitian operators• Eigenvalue equations• Hermitian operators

– Real eigenvalues– Orthogonal eigenfunctions if eigenvalues different

• Eigenvalues only possible outcomes of measurement

• Spin-½ operators only represented by 2 x 2 matrices