Spin Dynamics Basic Theory Operators Richard Green SBD 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 entity Transforms one function into another i.e. operators only act on functions (incl. vectors) g = Ôf Unit Operator g = ˆ1 f means g = f
Example of operator action ψ 2 > g = Ô f > = Ôf > Ô f > ψ 1 >
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? O( ˆ α f+ β f ) = α(o ˆ f ) + β(o ˆ f ) 1 2 1 2 For all functions f 1 and f 2 ; and For all complex constants α and β
Not all operators linear LINEAR d Ô f ( x) = f ( x) dx d d d df1 df2 ( α f1( x) + β f2 ( x)) = ( α f1( x)) + ( β f2 ( x)) = α + β dx dx dx dx dx NON-LINEAR Ô f ( x) = e f ( x) e = e e αe + βe [ α f ( x) + β f ( x)] α f ( x) β f ( x) f ( x) f ( x) 1 1 2 1 2
Hermitian operators definition Hermitian operators satisfy this condition f Oˆ g = O ˆ f g VERY IMPORTANT For any normalisable functions f and g
Eigenfunctions, eigenvalues and eigenvalue equations Ôf =λ f Eigenvalue equation f = eigenfunction λ = eigenvalue (complex constant)
Eigenfunction basis sets An operator may have more than one eigenfunction and eigenvalue Of ˆ Of ˆ Of ˆ f 1 1 1 f 2 2 2 f 3 3 3 ˆ n = λ n f n Of = 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 pˆ = iħ( + + ) x y z Kinetic Energy 2 2 2 2 ˆ ħ = ( + + ) 2 2 2 2m x y z E kin Potential Energy V ˆ( x) = V ( x)
Eigenvalue equation example Time independent Schrödinger equation ħ + x y z x y z = Eψ 2m 2 2 [ V(,, )] ψ(,, ) E ħ = + 2m 2 2 eigenvalue of operator [ V( x, y, z)] ψ( x, y, z) = corresponding eigenfunction
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. IMPORTANT Generally, the eigenvalues of Ô are the only possible outcomes of a measurement of O.
Matrix representation of operators Square matrices Compare vectors and functions as column matrices Vector transformation by operator Ô: / f O 1 11 O12 O 1n f 1 / f O 2 21 O22 O 2n f 2 = / f O n n1 On 2 O nn f n
Orbital angular momentum operators For information only by way of comparison to what follows ˆ Lx= iħ y z z y Lˆ y= iħ z x x z Lˆ z= iħ x y y x Can be derived from classical expressions
Orbital angular momentum commutators Lˆ, Lˆ = i Lˆ ħ x y z Lˆ, Lˆ = i Lˆ ħ y z x Lˆ, Lˆ = i Lˆ ħ z x y
Spin (angular momentum) operators No classical starting point exists Approach Impose experimentally-observed constraints ± ħ 2 Impose need for spin operators to be linear and Hermitian (because a physical property) Assume commutation relationships similar to orbital angular momentum obeyed
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 n φ θ Sˆ n iφ ħ cosθ e sinθ = iφ 2 e sinθ cosθ y x
General representations for arbitrary direction n Spin eigenvectors iφ cos( θ / 2) e sin( θ / 2) = n and = iφ n e sin( θ / 2) cos( θ / 2) These two vectors provide orthonormal basis for spin space such that any spin state A> can be written: A = c + c 1 n 2 n
Pauli spin matrices used in NMR for the x, y and z directions θ θ θ 1 ˆ 0 1 = 90 ϕ= 0 I x= 2 1 0 ˆ 1 0 i = 90 ϕ= 90 I y= 2 i 0 1 ˆ 1 0 = 0 ϕ= 0 I z= 2 0 1
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