Download χ = a α+ b β

PHYS 30101 Quantum Mechanics
Lecture 14
Dr Gavin Smith
Nuclear Physics Group
These slides at: http://nuclear.ph.man.ac.uk/~jb/phys30101
Syllabus
1. Basics of quantum mechanics (QM)
Postulate, operators,
eigenvalues & eigenfunctions, orthogonality & completeness, time-dependent
Schrödinger equation, probabilistic interpretation, compatibility of
observables, the uncertainty principle.
2. 1-D QM Bound states, potential barriers, tunnelling phenomena.
3. Orbital angular momentum
Commutation relations, eigenvalues
of Lz and L2, explicit forms of Lz and L2 in spherical polar coordinates, spherical
harmonics Yl,m.
4. Spin
Noncommutativity of spin operators, ladder operators, Dirac notation,
Pauli spin matrices, the Stern-Gerlach experiment.
5. Addition of angular momentum
Total angular momentum
operators, eigenvalues and eigenfunctions of Jz and J2.
6. The hydrogen atom revisited
Spin-orbit coupling, fine structure,
Zeeman effect.
7. Perturbation theory
First-order perturbation theory for energy levels.
8. Conceptual problems
The EPR paradox, Bell’s inequalities.
4. Spin
4.1 Commutators, ladder operators, eigenfunctions, eigenvalues
4.2 Dirac notation (simple shorthand – useful for “spin” space)
4.3 Matrix representations in QM; Pauli spin matrices
4.4 Measurement of angular momentum components: the
Stern-Gerlach apparatus
RECAP: 4. Spin
(algebra almost identical to orbital angular momentum algebra –
except we can’t write down explicit analogues of spherical harmonics
for spin eigenfunctions)
Commutation relations
(plus two others by cyclic
permutation of x,y,z)
By convention we choose to work with eigenfunctions of S2 and Sz
which we label α and β
So, the eigenvalue
equations are:
Any general spin-1/2 wavefunction χ can be written as a linear
combination of the complete set of our chosen eigenfunction set:
χ = a α+ b β
(there’s only two
eigenfunctions in the set)
The coefficients a and b give the weighting and relative
phases of the α and β eigenstates.
Normalization: a2 + b2 = 1
The wavefunction χ could be, for example, that of a spin-1/2
particle polarised in the x-direction (an eigenstate of Sx)
We now find the coefficients a, b for this state as an example
Eigenfunctions and eigenvalues of Sx, Sy, Sz
described in this way:
χ = a α+ b β
RECAP: 4.2 Dirac notation
Dirac
4.3 Matrix representations in QM
We can describe any function as a linear combination
of our chosen set of eigenfunctions (our “basis”)
Substitute in the eigenvalue
equation for a general operator:
Gives:
4.3 Matrix representations in QM
We can describe any function as a linear combination
of our chosen set of eigenfunctions (our “basis”)
Substitute in the eigenvalue
equation for a general operator:
Equation (1)
Gives:
Multiply from
left and integrate:
)
(We use
And find:
Exactly the rule for
multiplying matrices!
Matrix representation:
Eigenvectors of Sx, Sy, Sz
Eigenfunctions of spin operators
(from lecture 13)