Quantum Mechanics

Front 1

1-dimensional Schrodinger Equation?

Back 1

Ĥψ = Eψ

Ĥ = -ħ²/2m.d²/dx² + V(x)

Front 2

How do we interpret a wavefunction?

Back 2

interpret as telling us the probability of the particle(s) being at that point in space: the square modulus of the wavefunction is proportional to the probability density

Front 3

eigenfunctions?

Back 3

solutions to the Schrodinger equation

Front 4

eigenvalues?

Back 4

corresponding energies E to solutions to the schrodinger equation

Front 5

for this course what do we need to work out the Hamiltionian for a system?

Back 5

The kinetic energy operator, potential energy as a function of the coordinates

Front 6

1 dimension operators?

Back 6

Front 7

3 dimensions operators?

Back 7

Front 8

What are postulates?

Back 8

basis of phisical laws that are not obvious in themselves, but when followed through describe nature as we observed it (with some approximations)

Front 9

Postulates of Quantum mechanics?

Back 9

1)The state of a system is fully described by its wavefunction

2)Each observable quantity (energy, position, momuntum, etc) is represented by an operator

3)if ψ is an eigen function of the operator for an observable, an attempt to measure the quantity will always give the corresponding eigenvalue, if it is not an eigenfunction then successive attempts to measure the quantity will give a random one of the eigenvalues

Front 10

expectation value of an operator?

Back 10

Front 11

probability of observing eigenvalue an? and if
ψ = ψn

Back 11

=>|∫ψ*ψndτ|²

ψ = ψn

= ∫ψ*A(hat)ψ dτ

=>A(hat)ψn = anψn

Front 12

Operator addition?

Back 12

Front 13

Back 13

Front 14

When are two operators equal?

Back 14

when they have the same effect on any possible operand

d/dx exp(x²) = 2x exp(x²)

but d/dx ≠ 2x because the operators would give different results on other functions

Front 15

Commutators?

Back 15

Front 16

[B,A] compared to [A,B]

Back 16

[B,A] = -[A,B]

therefore if [A,B] = 0, [A,B]=[B,A] therefore the two operators commute

Front 17

Linear operators?

Back 17

all operators encountered in QM are linear

A is linear if A(c₁f₁+c₂f₂)=c₁Af₁+c₂Af₂

=>if Af=af

=>A(cf) = cAf = caf= a(cf)

=>any constant times f is also an eigenfunction with the same eigenvalue a

Front 18

Normalisation?

Back 18

Eigenfunctions that differ only by a constant factor describe the same state so we choose a multiplying factor that makes manipulation as easy as possible: we normalise it

∫ψ*ψdτ = ∫|ψ²|dτ = 1

allowing us to treat ψ² as a probability densitiy

Front 19

How to normalise?

Back 19

Front 20

Harmonic Oscillator?

Back 20

Front 21

Classically forbidden?

Back 21

wavefunctions can tunnel into region where V(x) > E which is not allowed in classical mechanics

Front 22

Removing centre of mass motion?

Back 22

for a vibrating diatomic molecule with two atoms of mass m₁ and m₂

-Translational motion of the molecule with mass

M= m₁+m₂

-Vibrational motion of the nuclei as a single particle with reduced mass µ = (m₁m₂)/(m₁+m₂)

kinetic energy operator T(hat) = -h(bar)²/2µ d²/dx²

Front 23

guessed functional form of harmonic oscillator?

Back 23

Front 24

solve for the guessed functional form?

Back 24

Front 25

Reduced units?

Back 25

Front 26

raising operator?

Back 26

need operator such that [H,A] = A

[H,A] = HA -AH

consider H(Aψ) = (AH+[H,A])ψ = AHψ + Aψ

=AEψ+Aψ = (E+1)(Aψ)

A = raising operator - allows us to generate a new eigenfunction of H with a larger eigenvalue

Front 27

General energy level expression for harmonic osciator?

Back 27

Front 28

Exactly soluble problems in QM?

Back 28

Front 29

Approximate mechods in QM?

Back 29

Front 30

Basis set?

Back 30

functions that we know before we start, chosen on physical grounds to give the flexibility we think is needed

Front 31

transpose?

Back 31

Front 32

Kronecker delta notation?

Back 32

Front 33

Dirac notation?

Back 33

<m|Ahat|n>

imply integration over all space and spin coordinates <m|n> is the overlap integral

-If we write a state vector |n>, we imply that it is normalised

Front 34

expectation value?

Back 34

is the average value of a quantum-mechanical observable

<n|Ahat|n> abbreviated to <Ahat>n

Front 35

Hermitian operators?

Back 35

opertator for which <m|Ahat|n> = <n|Ahat|m>*

=>swapping over the two functions merely changes a matrix into its own complex conjugate

=> operators corresponding to observable quantities (including the Hamiltonian) are always Hermitian

Front 36

Two important properties of Hermitian operators?

Back 36

Their eignevalues are real

Eigenfunctions corresponding to different eigenvalues are orthogonal

Front 37

Variation principle?

Back 37

The expectation value of the Hamiltonian calculated using an approximate wavefunction is always greater than or equal to the true ground state energy of the system

Front 38

linear variatinoal method?

Back 38

write the trial wavefunction as a linear combination of basis functions, and seek of coefficients that minimize expectation value of H(hat)

Front 39

Functional variation approach?

Back 39

trial wavefunction is written with parameters and the parameters are varied to find the lowest and best energy

Front 40

Heisenberg Uncertainty principle?

Back 40

∆A∆B ≥ ½|<[Ahat,Bhat]>|

Front 41

Conserved quantities?

Back 41

commute with the Hamiltonian

-Orbital angular momentum (sometimes)

-spin angular momentum (often)

-Total angular momentum (always)

use these to define quantum numbers

Front 42

magnitude of orbital angular momenum?

Back 42

Front 43

Symmetric and antisymmetric spin functions?

Back 43

represent triplet and singlet states respectively

Front 44

Pauli principle?

Back 44

total wavefunction of a 2 eletron system is a product of its spatial and spin parts

-the total wavefunction is always antisymmetric with respect to exchange of identical fermions and symmetric with respect to exchange of identical bosons

Front 45

Fermions?

Back 45

Electrons and other particles with half-integer spins such as odd-mass nuclei

Front 46

Bosons?

Back 46

photons and other particles with integer spins such as even spins are bosons

Front 47

what space functions must a singlet spin function be paired with?

Back 47

singlet = antisymmetric

space = symmetric

Front 48

Hunds first rule?

Back 48

when two atomic or molecular states arise from the same electron configuration, the one with higher spin multiplicity is lower in energy

Front 49

explain Hunds first rule?

Back 49

since there is a coulomb repulsion between electrons, keeping them apart lowers the energy

conversely, two electrons with spins paired tend to pile together raising the energy

Front 50

Component of l in z direction?

Back 50

hbar ml

Front 51

number of values of ml

Back 51

for each l there are 2l +1 allowed values of ml

0, ±1,±2...±l