Quantum Year 2

What is a differential operator?

What is the Hamiltonian function and what does it represent?

H = T+V

Represents total energy of a system

What is the equation obtained by substituting H = V+T into TDSE?

What are two general properties of the Time Dependent Schrödinger Equation?

PDE - 1st order in time

PDE - 2nd order in space

What does it mean that the TDSE is linear in Ψ(x, t)?

If Ψ(x, t) is a solution of the Schrödinger equation, then aΨ(x, t) is also a solution where a is some complex number

If Ψ1(x, t) and Ψ2(x, t) are both some solutions then aΨ1(x, t) + bΨ2(x, t) where a and b are both complex numbers

What does the wave function describe?

The quantum state of the system

What does postulate probabilistic interpretation of Ψ via the Born rule give? + what is the equation

probability density for finding a particle at position x at time t

What is the equation for the probability of finding a particle between x and x+dx at a time t?

How do you find the probability that a particle is between x₀ and x₁?

Calculate integral

What are three constraints on the wave function?

Ψ(x, t) must be square-integrable

Ψ(x, t) must be continuous and single valued otherwise the probability is ill-defined

(equation in image) must be continuous

Explain the constraint that the equation in the image must be continuous

Otherwise there will be infinite momenta

Except when there is infinite discontinuity in V(x, t) (so only have this constraint when there is a finite discontinuity in the potential)

When is the potential of a system independent of time?

In closed systems, no external force

What is the general solution for the TDSE when the potential is independent of time?

What is the equation for the Time Independent Schrödinger Equation?

Explain what is meant by TISE is an eigenvalue equation

E is the energy eigenvalue

E represents total energy of the system

E is conserved for time independent potentials

What are stationary states?

TDSE solutions of the form shown in the image, also called energy eigenstates

What is the expression for E in the TISE?

What is the equation for E in the TISE for a free non-relativistic particle?

For a free particle with energy described by QM, is energy quantised or not quantised? Why?

NOT quantised because idk i haven’t checked with dan

What is the equation for the full wave function for a planar wave moving either in the positive or negative x direction?

For a particle in an infinite potential well, what does classical physics predict?

Continuous values of total energy, E=>0

What are the boundary conditions for the IPW?

ψ must be continuous - this means that ψ = 0 at the boundaries of the well

• x=0: Asink0+Bcosk0 gives B=0

• x=L: AsinkL=0 gives kL=nπ, n=1,2,3..

Solutions are discrete (quantised) states q

What is the wave function for a particle in the IPW?

What is the equation for k_n in the IPW?

What is the associated momentum for a particle in the IPW?

What is the total energy associated with a particle in the IPW?

When normalising IPW wave functions, what is the equation for A?

When normalising IPW wave functions, what is the full equation for the wave function?

What does HUP stand for?

Heisenberg Uncertainty Principle

What is the HUP for position and momentum?

What are the two uncertainties associated with E_n from the HUP?

What is the equation for the number of nodes for the n^th state?

num nodes = n-1

What is the equation for ψ’(x)?

Why is it allowed for ψ’(x) to not be continuous at x=0?

There is an infinite jump in V(x) at x=0,L

What are the conditions for the symmetric IPW?

How do you obtain wave functions from the symmetric IPW?

Taking the solutions of the IPW at 0<x<L and making a substitution x→x+L/2

What are two pieces of information gained by solving the symmetric IPW?

odd n, symmetric, even parity

even n, antisymmetric, odd parity

What does parity relate to?

symmetry, transformations about the y-axis

What is a parity transformation?

Reflecting in the y-axis

What does antisymmetric mean?

Odd symmetry

Draw a diagram for the FPW with the conditions on the potential

Including E and KE

In the FPW, when E>V₀, what is happening?

Unbound states

Total energy E continuous, not quantised

In the FPW, when E<V₀, what is happening?

Bound states

Expect discrete values

In Region I (-L/2<x<L/2) of the FPW, what are the solutions for the wavefunction?

ψ1(x) = A sin kx + B cos kx 

In Region II (x>L/2) of the FPW, what are the solutions for the wavefunction?

ψ2(x)= C exp(-αx) + D exp(αx)

with alpha² expression in the image

Set D=0 or not square integrable (blows up at large +ve x)

In Region II (x<-L/2) of the FPW, what are the solutions for the wavefunction?

ψ3(x)= F exp(-αx) + G exp(αx)

with alpha² expression in the image

Set F=0 or not square integrable (blows up at large -ve x)

For the FPW, in what way is there symmetry and what should you expect as a result?

potential is symmetric wrt x=0 -> expect symmetric and antisymmetric states (odd and even parity)

Considering even-parity solutions for the FPW only, what happens for larger and smaller values of V₀?

Larger V₀ gives more bound states

Smaller V₀ gives fewer bound states

In the FPW what is the condition for the number of bound states?

There is always at least one bound state, even in a very shallow well (V₀→0)

Compare the wavenumber and energy of the nth state between the IPW and the FPW

the wavenumber and energy of the nth state is always less in the FPW than in the IPW

For even-parity only in the FPW, when the condition in the image is true, how many symmetric or antisymmetric states are there?

Only one symmetric state exists

For odd-parity only in the FPW, when the condition in the image is true, how many symmetric or antisymmetric states are there?

No antisymmetric state exists

Compare the FPW and the IPW

Infinite well: 

• ψ(x) confined to the well

• k_n = nπ/L

• Infinite tower of states

• No unbound states

Finite well:

• ψ(x) spreads out beyond the well

• k_n and energies lower

• Finite tower of states

• Unbound states when E>V₀

Why are the energy levels in the FPW lower than in the IPW?

Lower in the FPW because the wavefunction spreads out by penetrating the classically forbidden region, and reduces its kinetic energy

In the FPW, at x>L/2, what is the wavefunction proportional to?

e^-αx

In the FPW, at x<-L/2, what is the wavefunction proportional to?

e^αx

What is the expression for α² in the FPW?

As V₀→infinity, what happens to α and 1/α in the potential well?

α→infinity

1/α→0

As E→V₀, what happens to α and 1/α in the potential well?

α →0

1/α→infinity

What is the definition of penetration depth?

The depth of quantum tunnelling

What is penetration depth or depth of tunnelling determined by?

1/α

What does non-zero wavefunction in classically forbidden regions allow?

Tunnelling between classically allowed regions

What are quantum energy states a typical property of?

Any well-type potential

Describe non-zero tails

Corresponding wave functions and probabilities for quantum energy states are mostly confined in the potential well but exhibit non-zero tails in the classically forbidden regions of KE<0

When are the non-zero tails in the classically forbidden KE<0 region not allowed?

When V(x)→infinity

What are 2 consequences of the Heisenberg Uncertainty Principle in the context of quantum states in potential wells?

The lowest energy state is always above the bottom of the potential and is symmetric

The wide and/or more shallow the potential, the lower the energies of the quantum states

Inside the FPW is the number of quantum states finite or infinite?

Finite

In a potential well, when the total E is larger than the height of the potential, is the energy continuous or quantised?

Continuous

When V=V(x) in the potential well, what does form do the time-dependent wave functions take and what does this mean?

This means that both the bound and continuous states are stationary

In the example {λ_n} what do the curly brackets denote?

The set of

What does orthonormal mean?

The function is both orthogonal and normalised

What does it mean to say “the different potentials V(x) give different Hamiltonians for which there are different sets of energy eigenstates”

Different potentials lead to different sets of energy levels

What is the eigenvalue equation involving the linear operator Ô?

{λ_n} and eigenfunctions {u_n(x)} are solutions

What is δ_nm and what does it mean?

Kronecker delta

δ=1 when n=m, δ=0 when n!=m

What is the expression for the eigenfunctions corresponding to different eigenvalues λ_n = λ_m?

The set of eigenfunctions {u_n(x)} form a complete set in a space of functions: how can any function u(x) that obeys the same boundary conditions as the u_n(x) be expressed?

What does a_n represent in this expression?

the coefficients of expansion, probability amplitudes

What is an observable?

A measurable quantity

What is the operator for position x?

x^

What is the operator for momentum?

What is the operator for kinetic energy?

What is the operator for potential energy?

What is the operator for total energy E?

What is the operator for parity?

Why do bound states have a definite value of momentum squared but not of momentum?

Because do not know direction which defines whether it is the positive or negative root

What are the two solutions for parity eigenfunction and what do they refer to?

What is the postulate for QM that refers to the probability of measuring a specific eigenstate?

the system described by ψ(x) can be found in the state ψ_n(x) with probability

P_n = |a_n|²

What does collapse of the wavefunction refer to?

When a measurement has been made yielding E_n, the system collapses to the state ψ_n(x) and remains there

i.e. the act of measuring the system changes the wavefunction

What is the integral equation for the associated average value for some observable λ at a specific time t?

What does the equation for the average value of an observable become if we know the probability amplitudes a_n?

What are the two equations for the rms fluctuation?

What does the rms fluctuation show?

The spread of values around the average value of the observable <λ>

What does ∆λ provide information on?

How much a measured value of λ can differ from the average value <λ>

What is the value of ∆λ when the identical systems are all in an eigenstate ψ_n of some linear operator Ô associated with an observable λ?

What is the condition for when two observables can be measured simultaneously with absolute precision?

When the commutator of the associated operators of two observables A and B, [Â , B^]=0, then the observables can be measured simultaneously

When can two observables A and B be measured with total precision?

When the commutator [Â , B^]=0

What is the equation for the commutator of two operators  and B^?

Why can two observables A and B be measured with total precision when the commutator of their associated hermitian operators equals 0?

If [A^,B^]=0 then A^ and B^ have a common set of eigenfunctions  

The system is initially at  

The measurement of A makes the system collapse 

This yields an eigenstate ψ_n, A_n

Since A^ and B^ have a common set of eigenfunctions, this means that a measurement of B yields B_n, the system remains in the state ψ_n, and ∆B=0

What is the condition for when two observables CANNOT be measured with total precision?

What is the full HUP equation?