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? (p)

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 (not subject to any outside forces so V = 0)

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 (maximises fermi holes so electrons less likely to be close)

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] psi

(AB - BA) psi

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>

result of ^s₁z α

½ h- (ms h-)

eigen value for ^Sz

Ms h-

eigenvalue for ^Lz

M_L h-

add to psi total to make it antisymmetric for spin

α1β2 - β1α2

n for ground state En

1

considerations when 2 wavefunctions (x and y) have an energy

they dont have to have the same n

energy units of eigenvalue

Joules

[x, p] =

i(h-)

[Jx, Jy] =

i(h-) Jz

[Jz, Jx²]

i(h-) [JxJy + JyJx]

[Jz, J²] =

0

How can AM² be meaured accurately

• only 1 of Jx, Jy, Jz can be measured

• convention is Jz

  • can only measure AM² accurately in Z direction

quotient rule for f(x) = f/g

d/dx = (f’g - g’f) / g²

variation principle

• E » E0

• E ~ <ψ* | H | ψ> / <ψ* | ψ >

• minimum E is found by dE/dα

if <H> = -1/(2r²) d/dr r² d/dr - 1/r, what is <T>?

-1/(2r²) d/dr r² d/dr

if <H> = 1/(2r²) d/dr r² d/dr - 1/r, what is <V>?

-1/r