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Spin Dynamics Basic Theory Operatorsv2 - … Dynamics Operators.pdf · Spin Dynamics Basic Theory...
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Transcript of Spin Dynamics Basic Theory Operatorsv2 - … Dynamics Operators.pdf · Spin Dynamics Basic Theory...
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
=
=
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