Quantum mechanics

what is h- “h bar” in terms of h?

h/2π

If psi is normalised, what is ∫psi*psi dx

1

What is the expectation value, <x>, for un-normalised psi(n)?

(∫psi* x psi d(tao)) / (∫psi* psi d(tao))

what is the uncertainty of x, ∆x?

∆x = √(<x²> - <x>²)

V^(x)

1/2(kx²)

p^ x

-i (h-) d/dx

Heisenberg uncertainty principle

∆x ∆p >= ½ (h-)

general eigenfunction/value equation

Q^ psi(q) = q psi(q)

What is the uncertainty of Q, ∆Q?

√(<Q²> - <Q>²)

Hamiltonian eigenvalue equation

H psi = E psi

H =

T + V

What is T(x) eqn?

Kinetic energy, p²/2m

Psi must be a _ function

continuous

momentum of a QM particle

definite (p^ psi = p psi)

Hamiltonian of a free particle

T^ (kinetic energy)

PIB (particle in a box) assumptions

V^ (free energy) =0, psi(0) = psi(a) = 0

Result of boundary conditions

quantisation

Test if 2 waves are orthogonal

∫psi(m)*psi(n) dx = 0

Hamiltonian eqn for PIB

-(h-)²/2m d²/dx² (kinetic energy)

Integreal limits for PIB

a, 0

general form of a PIB eigenfunction

A sin(px/(h-)) + B cos(px/(h-))

role of A in the eigenfunction for PIB

Normalisation constant (determines probability density)

Reason for quantised E in PIB

eigenvalue depends on n which can only take certain values

probability of finding e- in spherical shell

∫∫ R(r )² Y(θ, phi)² dV (R(r ) and r²dr are constants)

V^ in au (atomic units)

-Z/r (hartrees)

Z for H atom

1

Pauli (exclusion) principle

no two electrons in an atom can have the same set of four quantum numbers / wavefunction

Hunds 1st rule

lowest energy is given by the maximum possible S, spin quantum number

Hunds 2nd rule

maximise L subject to S for the lowest energy

Hunds 3rd rule

< half full: lowest J is the most stable

> half full: highest J is most stable

[A, B]

AB - BA

uncertainty principle for [A, B]

∆A∆B >= ½ |<[A,B]>|

delx (x) = (operator identity, del = partial derivative)

1 + x delx

[Jz, Jx²] =

JzJxJx - JxJzJx + JxJzJx - JxJxJz

[Jz, Jz²] =

0 (AM operators)

asymmetrical / triplet spin function

1/√2 (a(1)b(2) - a(2)b(1) )

symmetrical /singlet spin function

1/√2 ( a(1)b(2) + a(2)b(1) )

2px in |nlm>

1/√2 (|-211> + |21-1>)

2py in |nlm>

i/√2 (|211> + |21-1>)

2pz in |nlm>

|210>