Quantum

Question 1

Time dependent wave fucntion

Answer 1

corresponds to a single -particle system, of mass m and position r

Question 2

eigen values of a hermitian operator are ..

Answer 2

real and orthonormal

Question 3

eigen values of a hermitian operator have eigenfunctions that are

Answer 3

orthonormal and form a complete set

Question 4

The expansion postulate

Answer 4

a system will not n general be in one of the eigenstates phin, but in a more general state (wave function symbol) which is a linear combination of these basis functions eigenstates forming a complete set

Question 5

when are they compatible

Answer 5

if they share the same set of eigenfunctions

Question 6

commute … –>

Answer 6

if the commute, they share eigenstate, can be measured simultaneously and are compatible

Question 7

L^2 =

Answer 7

Lx^2 + Ly^2 + Lz^2

Question 8

assumption of time-independent perturbation theory works as an approximate method.

Answer 8

i) The pertubation is small to a system with known eigen solutions, such that
H^ = H^0 + BH^
hamiltonian of the pertubated system,= hamiltonian of the known system + pertubation hamiltonian B= pertubation strength

ii) the eigen energy is expanded such that,
En = Eon + BE1n +B^2 E2n + …

iii) the wavefunction is expanded such that
Un= U0n + BU1n + B^2U2n + …

iv) the correction terms can be found by known the
H^oU0n = E0nU0n eigensolutions & forms of H^’

Question 9

Perturbation theory H’

Answer 9

H’ is the perturbation Hamiltonian assumed to be a small perturbation to the known system, e.g as shown on diagram would be a potential spike at x=x0

Question 10

Perturbation Theory E0n

Answer 10

E0n are the eigenenergy solutions corresponding to the nth state for the unperturbed system, which are also known

Question 11

Perturbation Theory E1n

Answer 11

E1n is the 1st order, eigenenergy correction corresponding to the nth solutions and is calculated from the eigen expectation value obtained from the effect of H’ acting on the nth unperturbed state U0n

Question 12

Perturbation Theory U0n

Answer 12

U0n are the unperturbed wavefunctions with known solutions

Question 13

properties of the hamiltonian

Answer 13

the hamiltonian can be applied to hydrogenic-type atoms e.g. H, He+, Li2+ etc
it comprises of 3 terms, the first relates to translational KE the 2nd to torational KE and the 3rd to the potential energy
it is derived assuming the central force theroem and that the atomic system has a spherically symmetric potential

Question 14

what are alphaz and beta z

Answer 14

alphaz and beta z are the spin up and spin down spins tates quantized in the z direction

Question 15

commutators

Answer 15

determine the compatibility of two measurements
if [A,B]=0 measurements are compatible and A & B have simultaneous eigenfunctions
if [A,B] does not =- then a second measurement renders the first invalid as A & B do not have simultaneous eigenstates
in the case [L^2 ,Lxyz] =0 this is true because L^2 =Lx^2 + Ly^2 + Lz^2 making these by definition compatible operators with shared eigenstates

Question 16

expansion postualte

Answer 16

wavefunction = sum ai thi i 
wav function = 
[a1
a2
.
an]
wave function ^+ = sum [a1*, a2* ...an*]

Question 17

normalisation condition

Answer 17

wavefunction+wavefunction =1
therefroe
sum |a1|^2 =1

Question 18

the first order of perturbation correction

Answer 18

the first order of perturbation correction shifts the energy eigenvalues of the unperturbed system (Eon) by
= E1n , which is the expectation value associated with H’ acting on the unperturbed Uon state

Question 19

by inspection show that matrix operators are hermitian

Answer 19

hermitian operators in matrix form, in genera, have matrix elements Ax,y =A*y,x. where x does not equal y
this is satisfied in each case for sx sy and sz