Simple Quantum Mechanical Systems

Question 1

Time independent Schrödinger equation with Hamiltonian operator plugged in

Answer 1

Question 2

Derive time independent Schrödinger equation with Hamiltonian plugged in

Answer 2

Question 3

Factorise Ψ(x,t) for separation of variables

Answer 3

Question 4

variable separated time independent Schrödinger equation? significance?

Answer 4

LHS Depends only on t, RHS depends only on x

Question 5

Solution to LHS of variable separated time independent Schrödinger equation

Answer 5

Question 6

How to form time independent Schrödinger equation with Hamiltonian plugged in

Answer 6

-factorise Ψ(x,t) = Φ(t)Ψ^~(x)

-separate variables

(107 is RHS of variable separated time independent Schrödinger equations)

Question 7

What does time independent Schrödinger equation with Hamiltonian plugged in suggest ? (What else does this mean)

Answer 7

Ψ^~(x) is an EIGENFUNCTION of Hamiltonian Operator

A particle in state Ψ^~(x) has definite energy value = E

Question 8

Time dependent wave function

Answer 8

Where Ψ(x,t) = Φ(t)Ψ^~(x)

Question 9

General solution of time independent Schrödinger equation (particle in a box)

Answer 9

Question 10

Answer 10

Ψ(x,t) is localised (t=0)

Question 11

Time independent Schrödinger equation with non trivial potential takes what form w Hamiltonian operator

Answer 11

Question 12

1d Particle in a box;
State assumptions

Answer 12

Particle moves along 0<x<L
Outside of this wave function vanishes

Question 13

1d Particle in a box;
General solution to time independent Schrödinger equation with Hamiltonian plugged in

Answer 13

Question 14

1d Particle in a box;
Determine Energy of particle

Answer 14

Question 15

1d Particle in a box;
Normalise wave function

Answer 15

Given Ψ(x) = Acos(kx) +Bsin(kx)
=> |Ψ(x)|^2 = B^2(Sin(kx))^2
=>

Question 16

1d Particle in a box;
Form time dependent solutions to Schrödinger equation
Then use to give corresponding probability densities

Answer 16

Where
E_n = (π^2h^-2n^2)/(2mL^2)
gen solution is
Ψ(x) = Acos(kx) +Bsin(kx)
=> |Ψ(x)|^2 = B^2(Sin(kx))^2
And B = sqrt(2/L)
=>

Question 17

Relate Ψ_loc to Ψ_n

Answer 17

Ψ_loc is a state that is a superposition of many eigenstates Ψ_n

Question 18

1D particle in a box;
Decompose eigenstates of Ψ_loc (x)

Answer 18

Question 19

1D particle in a box;
Decompose eigenstates of time dependent Schrödinger equation

Answer 19

(121) is solutions to time dependent Schrödinger equation
(124) is decomposition of localisation of time independent wave function

Question 20

1D particle in a box;
Define a potential well

Answer 20

Question 21

1D particle in a box, potential in a well;
Solve portion where potential is negative
What are assumptions ?

What is form of solution

Answer 21

Question 22

1D particle in a box, potential in a well;
Even case

Answer 22

Question 23

1D particle in a box, potential in a well;
Odd case

Answer 23

Question 24

1D particle in a box, potential in a well;
Determine energy levels when -U_0 < E <
0

Answer 24

Question 25

1D particle in a box, potential in a well; What happens if E<= -U_0 ?

Answer 25

Question 26

Derive uncertainty of a general operator Ο^^ I.e. ΔΟ^2 And prove that this is equal to the expected squared derivation of Ο^^ from the expected value <Ο^^> Ie

Answer 26

Question 27

1D particle in a box, potential in a well, E > 0; First define V_0 And what it means

Answer 27

V_0 = -U_0 For V_0 this is a potential barrier If E < V_0 then the classical particle would get reflected

Question 28

1D particle in a box, potential in a well, E > 0; Define U(x) for 3 regions of potential well

Answer 28

Question 29

1D particle in a box, potential in a well, E > 0; Solve time independent Schrödinger equation for region 1

Answer 29

Question 30

1D potential barrier, E > 0; Solve time independent Schrödinger equation for region 2

Answer 30

Question 31

1D particle in a box, potential barrier, E > 0; Solve time independent Schrödinger equation for region 3

Answer 31

Same as region 1 with different coefficients

Question 32

1D particle in a box, potential barrier, E > 0; Full time dependent solution for a given p

Answer 32

Question 33

Wave packet? Called this why?

Answer 33

Ψ(x,t) for U(x) = 0 At t=0 localised near x=x_0 with uncertainty Δx and momentum localised around p Oscillates in x, hence wave Localised, hence packet

Question 34

As time goes on, wave packet described by Ψ(x,t) moves to

Answer 34

Question 35

Relate solution to wave packet to plane waves

Answer 35

p->q and the wave is no longer localised/idealised

Question 36

Construct relation between free particle moving through potential barrier

Answer 36

Question 37

Ψ(x,t)_scattering?

Answer 37

Choosing x_0 << -L/2 so that Ψ(x,t) is localised entirely for x < -L/2 at t=0

Question 38

Quantum tunelling?

Answer 38

Quantum particle can penetrate a potential barrier even if it has energy that is lower than the height of the barrier

Question 39

1D particle in a box, potential in a well, E > 0; Tunnelling probability is small if

Answer 39

Question 40

Relate R(p) and T(p)

Answer 40

Add to equal 1

Question 41

When is R(p) zero

Answer 41

When sin(p₂L/h)=0

Question 42

One deBroglie wavelength

Answer 42

h/p Where h is non reduced p.constant

Question 43

1D particle in a box, potential in a well, E > 0; U(x) has jump discontinuity where?

Answer 43

x= +-L/2

Question 44

General solution to LHS of variable separated Schrödinger equation (particle in a box)

Answer 44

Question 45

General solutions to RHS of Schrödinger equation (particle in a box)

Answer 45

Impose constraints if particle in box to find constants , e.g: Ψ(0) = Ψ(L) = 0

Question 46

What is the meaning of ψ localised

Answer 46

This is a approximation to a classical situation The **P** density spikes in a small region and is almost zero everywhere else

Question 47

Explain superposition

Answer 47

A particle can exist in many states A localisation is the sum of these states that forms a wave packet

Question 48

Scattering amplitudes?

Answer 48

A+ and C+

Question 49

What is this? What is their important property?

Answer 49

Coefficients of Fourier transform of Ψ(x,0) They are Sharply peaked in abs value near q~p which reflects the fact that Ψ(x,t) describes a particle whose momentum is approximately p with non zero uncertainty of P

Question 50

When calculating Ψ(x,t)_scattering remember

Answer 50

We can replace -inf in integral bound with zero as area is negligible due to localisation

Question 51

How to choose

Answer 51

It is chosen such that (below) is equal to Ψ(x,t) where Ψ_q(x,t) is the plane wave

Question 52

Ψ_scattering(x,t) for region 1?

Answer 52

Question 53

Ψ_scattering(x,t) for region 3?

Answer 53

Question 54

Limitation to potential barrier

Answer 54

Modelled as instant jump, in practise would be a more shallow gradient