Section 2

Question 1

Answer 1

h/2π (reduced Planck constant)

Question 2

A wave that describes an electron should have

Answer 2

And p=mv

Question 3

Simple, planewave

Answer 3

Question 4

Energy of free electron in empty space

Answer 4

Question 5

(Time dependent) Schrödinger equation

Answer 5

Simplest equation to describe free plane wave

Question 6

Answer 6

Question 7

Answer 7

Question 8

Schrodinger’s equation of motion under a potential

Answer 8

Question 9

Answer 9

-(e^2)/x

Question 10

Answer 10

Question 11

Answer 11

Quantum state of particle

Question 12

Answer 12

Question 13

Answer 13

Sqrt(2pi)

Question 14

Normalisable, functions have

Answer 14

Question 15

Answer 15

Question 16

Answer 16

Question 17

Answer 17

Question 18

At a time t, uncertainty of position of plane wave approximated by

Answer 18

Question 19

Heisenberg’s uncertainty, principle

Answer 19

Question 20

Free Schrödinger equation can be used to determine

Answer 20

Wave function for all t starting with t=0 and

Question 21

Probability density of a wave function is

Answer 21

Gaussian and centred at x^-_0(t)

Question 22

The centre of the probability density of a wave function moves with?

Answer 22

Velocity (classical velocity of particle)

Question 23

For large times, how to deal with uncertainty of x

Answer 23

Ignore initial uncertainty and let

Question 24

After a large time t, particles coordinates x_i wil most likely be in

Answer 24

(That p_1 should be p_i)

Question 25

Linear operator for momentum

Answer 25

Question 26

Let p^{^} _i act on the wave function of a plane wave

Answer 26

(As this wave a definite momentum p^- ψ is just multiplied by the value of p_i

Question 27

A wave function ψ is an eigenfunction function of Ο^{^} if ? What does that mean then?

Answer 27

With eigenvalue Ο€ **R** This means that the observable described by Ο^{^} has a definite value in the state described by ψ

Question 28

Expectation of Ο^{^} in a normalised state given

Answer 28

Where Ο^{^} is general observable

Question 29

Rewrite Schrödinger equation using operators

Answer 29

- use momentum linear operator (63)

Question 30

What does the Hamiltonian operator do

Answer 30

Generalises the classical expression below

Question 31

What is the Hamiltonian operator (equation)

Answer 31

Question 32

Hermitian operator

Answer 32

Question 33

Prove momentum operator is Hermitian

Answer 33

Integration by parts to get 2nd last line

Question 34

Compute time derivative of normalisation condition

Answer 34

Given that the Hamiltonian (H^{^}) is a Hermitian operator And Ψ_1 = Ψ_2 = Ψ Therefore last line = 0 Therefore norm is constant

Question 35

Conservation of norm is what?

Answer 35

As the norm differentiates wrt time to give zero Norm is constant Therefore if norm = 1 at one moment of time It is 1 for all t Total probability is conserved under time evolution <=> time evolution is unitary

Question 36

What is the commutator

Answer 36

Question 37

For any observable operator, the time derivative of its expectation value is given by

Answer 37

Question 38

Prove result of time derivative of expectation of any observable operator

Answer 38

2.2 is definition of Hermitian Sub in Hamiltonian for time derivative

Question 39

Commutator of Hamiltonian operator and momentum operator

Answer 39

Question 40

How to solve commutator of the Hamiltonian operator and the momentum operator

Answer 40

-use linearity to separate into (below) -

Question 41

Compute commutator of potential of position and momentum operator

Answer 41

Where (63) is momentum operator = -ih(d/dx_i) (line through h)

Question 42

Compute commutator of momentum from Hamiltonian and momentum operator

Answer 42

Question 43

Combined solution to commutator of Hamiltonian operator and momentum operator

Answer 43

(87) is [H^{^}, p^{^}_i] = 1/(2m) [p^{^2}, p^{^}_i] + [U(x^-), p^{^}_i] (91) is [U(x^-), p^{^}_i] = i**h**d(U(x^-))/dx_i

Question 44

Time derivative of momentum operator in any state

Answer 44

Using commutator of Hamiltonian (first line) (2.3) formula for time derivative of expectation of any observable operator

Question 45

[H^{^}, x^{^}_i] = ?

Answer 45

Question 46

Time derivative of expectation of position operator

Answer 46

Question 47

What are Ehrenfest Theorem?

Answer 47

Quantum contrast to classical equations of motion (classical below)

Question 48

The Ehrenfest Theorem

Answer 48

Just top and bottom equations (middle used to derive bottom)

Question 49

To integrate Gaussian function

Answer 49

Sqrt(π*σ)

Question 50

Answer 50

Question 51

Process when normalising wave

Answer 51

1) integrate abs val 2) let’s abs val equal 1 3) rearrange to isolate A

Question 52

Normalising wave in R³

Answer 52

When changing variables to change dx³ to dx just cube entire integral

Question 53

What is this

Answer 53

centre of Gaussian distribution of position

Question 54

What is the wave function of a particle likely to be found at t=0 near x=x_0

Answer 54

Question 55

What is an operator?

Answer 55

Linear map from vector space to self O: V -> V

Question 56

Prove time Unitarian

Answer 56

Using Schrödinger equation at top

Question 57

3 facts about Hermitian operators

Answer 57

1) sum of Hermitian operators is Hermitian 2) product of Hermitian operators is Hermitian 3) all R is Hermitian