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Quantum Flashcards

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1
Q

State TDSE.

A
  • μ = reduced mass
    • usually just mass of the particle
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2
Q

State TISE.

A
  • μ = reduced mass
  • usually just mass of the particle
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3
Q

How do you obtain TISE from TDSE? What is the physical interpretation of constant E?

A
  • Ψ(x,t) = T(t)ψ(x)
  • separation of variables
  • each indt term equals E
    • both constants have to be equal
  • E = total energy = sum of KE + PE
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4
Q

What is the time soln to TDSE?

A

T = e-iEnt/ћ

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5
Q

What is the significance of |Ψ(x,t)|2?

A
  • probability of finding the particle at posn x
  • stationary states => |Ψ(x,t)|2 associated with the wavefn of the state is indt of time
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6
Q

When is |Ψ(x,t)|2 expected to be indt of time (i.e. a stationary state)?

A
  • When the potential is indt of time
    • only then can the wavefn be split into time and space parts
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7
Q

Define the commutator between 2 operators A and B.

A

[A, B] = AB - BA

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8
Q

What are the implications of 2 operators if they commute?

A
  • compatible operators
  • so can be simultaneously measured
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9
Q

What is the angular momentum operator? Show commutation relation between operators for posn and momentum for particle moving along x is iħ

A
  • x =x & Px = iħ∂/∂x
  • [x, Px] = xPx - Pxx = iħ∂x/∂x = iħ
  • L is important for 3D problems involving central potentials (potentials that depend only on radial coordinate r)
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10
Q

What is the definition of a Hermitian operator? Give examples of Hermitian operators.

A
  • H = T + V
    • T = p2/2m = KE operator
    • V = V(x) = PE operator
    • What other examples are there?
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11
Q

Prove eigenvalues of Hermitian operator are real.

A
  • write eigenfn equations for the operator
    • also write the complex conjugate eigenfn equation for the complex conjugate operator
  • equate fns in definition with eigenfns from the equations above
  • evaluate LHS and RHS separately
    • show that qm = qm*
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12
Q

Explain the physical significance of n, l, m and give their possible values.

A
  • n = principal quantum number
    • energy => energy observable H
    • n = 1, 2, 3 …
  • l = angular momentum quantum number
    • magnitude of angular momentum
    • l = 0, 1, 2, …, n-1
    • corresponds to operator L2
  • m = magnetic quantum number
    • component of angular momentum along chosen axis
    • |m| less than or equal to l
      • -l , .., 0, …, l
    • corresponds to operator Lz
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13
Q

What is the notation 5f, and how many states correspond to these?

A
  • nl notation
    • n = 5
    • p => l = 3
      • so m = -3, -2, -1, 0, 1, 2, 3
        • giving 7 states
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14
Q

How to deduce L2 measured would be equal to an eigenvalue given the normalised angular wavefn?

A
  • L2 eigenfn equation:
    • L2 Y = l(l+1)ħ2 Y
  • find l by solving quadratic eqn
  • normalisation factor = an
    • probability given by |an|2

=> probability of getting eigenvalue l(l+1)ħ2 is |an|2 of Ylm corresponding to that eigenvalue

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15
Q

Work out expectation value of Lz given a normalised angular wavefun.

A
  • eigenvalues of Lz = mħ
  • expectation value of Lz = Σ |an|2mnħ
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16
Q

General form of solns to TISE for E > V0 in finite square well

A
  • same as that of a free particle
    • different wavenumber
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17
Q

General form of solns to TISE for a free particle.

A
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18
Q

What must the form of the equation be inside a potential barrier?

A

Exponential decay for the values of x

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19
Q

When is it more appropriate to use sin and cos instead of exponentials?

A
  • inside finite square well
  • inside infinite square well
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20
Q

Reflection probability

A
  • reflection coefficient = sqrt(R)
  • reflection probability = R
    • = |reflected flux/incident flux|
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21
Q

Transmission probability

A
  • transmission coefficient = sqrt(T)
  • transmission probability = T
    • = |transmitted flux/incident flux|
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22
Q

What is the general form of the wavefn in a periodic array of delta fn potentials?

A
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23
Q

What is the periodic boundary condition?

A
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24
Q

How can you deduce that a wavefn is an eigenfn of an operator?

A

If the wavefn is proportional to the operated wavefn

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25
Q

What are compatible operators?

A
  • if the operators commute
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26
Q

Quantum theory of measurement

A
  • λn = results obtained by measuring a physical quantity that corresponds to operator A
  • immediately after measurement, the system is in eigenstate (eigenfn) φn associated with eigenvalue λn returned from measurement
  • probability of measuring λn = |Cn|2
  • averages obtained after many measurements on identical particles = expectation value
    • ​expectation value of a quantum measurement is:
    • = integral of ψ*Aψ dT = Σn |Cn|2 λn
    • if the solns to TDSE are stationary
  • eigenfns are orthonormal if
    • integral of φn*φm dT = δm,n
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27
Q

What is meant by tunnelling in QM? Also why does it play an important role in nuclear fusion and alpha decay?

A
28
Q

What is meant by an orthonormal set of eigenfns?

A
  • eigenfns are orthogonal to each other
  • eigenfns are normalised
29
Q

State 3 conditions solns to the Schrodinger equation (SE), wavefns, must satisfy to be physically acceptable

A
  1. ψ & ψ’ must be continuous at boundaries
  2. normalisable (finite normalisation integral)
  3. single-valued
30
Q

Give an eigenvalue eqn other than SE.

A
  1. L2Y = l(l+1)ħ Y
    1. gives magnitude of angular momentum
  2. LzY = mħ Y
    1. gives component of angular momentum along a given axis
31
Q

What is the TISE for QHOs?

A
  • particle on a spring
    • => Hooke’s law with restoring force F=-kx
32
Q

Sketch ground state wavefn of QHO and give 2 ways in which the QHO differs from classical equivalent.

A
  1. energy spectrum is quantised
  2. minimum energy is not zero => zero point energy = E=(1/2)ħω

Exponential decay beyond classical limit

33
Q

Give the 3 approximations made when solving SE for periodic potentials?

A
  1. Born-Oppenheimer approx: Electron motion w.r.t fixed nuclei
  2. One electron approx: Each electron moves in a periodic potential due to fixed positive ions.
  3. 1D: Consider only one dimension, rather than 3.
34
Q

State Bloch’s theorem.

A

The eigenfunctions of the SE for electrons moving in periodic potentials is the product of the free particle plane wave and a function u(x) with the periodicity of the lattice. Wave functions of this form are called “Bloch wave functions”.

35
Q

What is the interpretation if the TDSE solution in terms of travelling waves?

A
36
Q

Which terms are neglected in the radial SE for r-> infinity and r-> 0?

A
  1. In the limit far from the nucleus, the effective potential term goes to zero and we have a free particle.
  2. Close to the nucleus, the angular momentum term l(l+1)/r^2 dominates in the potential.
    - 1/r term is the coulomb potential term and 1/r^2 is the centrifugal term.
37
Q

What is flux?

A
  • mean number of particles passing a point per unit time
38
Q

When the width of the square well is changed, how do we find the probability of occupying a given eigenstate?

A
  • Original wavefn not defined for new width, only for old width, e.g if changed from ±a, to ±3a, take coefficient integral from ±a, since fn defined for this range.
  • expansion postulate - find coefficient of eigenstate
  • Take square mod of coefficient.
39
Q

Explain why eigenfns of one operator might be eigenfns of another operator

A
  • if the operators commute
    • i.e. they are compatible
    • i.e. their commutator = 0
      • which means they can be measured simultaneously with certainty
      • and they share common eigenfns
40
Q

How do you obtain coefficients for above step potentials i.e. E>V0?

A
  • region before potential barrier
    • plane wave incident has amplitude 1
      • since assume a particle per unit length
      • travels to right so +ve x in exp
    • reflected at barrier is possible
      • reflected wave at barrier with reflection coefficient
      • travels to left so -ve x in exp
  • after potential barrier
    • transmission to another region
      • k changes to k’
    • no incident wave travels to the left
      • if d2ψ/dx2= 0
        • dψ/dx = C (const.)
        • ψ = Cx + D(another const.)
    • so coefficient with -ve x in exp = 0
41
Q

Obtaining K=ktan(ka)

A
  • match boundary conditions
  • find 2Acos(ka)
  • find 2kAsin(ka)
42
Q

What is an eigenfunction? What is an eigenvalue?

A
  • A function that when operated on returns itself & a multiplicative factor
  • The multiplicative factor is the eigenvalue
43
Q

What was deBroglie’s relationship between momentum and wavelength?

A

p=h/λ=ħk

44
Q

State the heisenberg uncertainty relation for position and momentum.

A
45
Q

Explain briefly why measuring an electrons position through compton scattering with high energy photon leads to uncertainty in the knowledge of the electrons momentum.

A
  • In scattering a photon of the electron, the uncertainty in the electrons position depends on the wavelength of the photon.
  • The smaller it is, the more accurately the position is known.
  • High energy photons have a higher momentum, and cause a larger change in momentum of the electron.
  • Therfore, the more accurately the position is known, the less accurately the momentum is known.
46
Q

Why can a wavefn be expressed as a Σnanφn?

A
  • eigenfns form a complete set meaning any fn satisfying same boundary conditions are linear combinations of total wavefn
47
Q

What are the possible values of spin quantum numbers s and ms for e-?

A
  • s = 1/2
    • ms = -1/2, +1/2
48
Q

What are the possible values of total angular momentum quantum number j that can be produced by combining l and s? What is the range of total magnetic quantum numbers mj?

A
  • |l-s| < j < |l+s|
    • in steps of one
    • those should be .. than or equal to
  • |mj| < j
    • in steps of one
49
Q

On which quantum number does energy depend and what is the functional form of this dependence?

A
  • Energy depends on n.
50
Q

What main compton scattering features contradict classical physics?

A
  • Upon scattering of particle, photon energy is reduced => wavelength increases.
  • Wavelength shift detected
  • photon momentum change
  • photon behaves as a particle
51
Q

Explain compton scattering.

A
  • beam of photons have associated momentum - hbar*k
  • electron momentum - mv
  • photon scatters from electron
  • total momentum and energy conserved Pfin=Pini
  • photon exchanges momentum with electron
  • electron recoils
  • photon momentum reduced
  • since photon speed invariant, wavelength must change
    *
52
Q

Physical interpretation of E_n.

A
  • En are the energies of the eigenstates.
  • Energy eigenvalue corresponding to nth eigenfunction.
53
Q

Define expectation value.

A
  • The average value of a set of repeated measurements on identical systems
54
Q

Expression for expectation of an operator in terms of of Cn and its eigenvalues.

A
55
Q

Describe the Stern-Gerlach experiment.

A
  • Produce beam of s-state atoms
  • l=0, so no orbital ang. mom. interaction with field
  • Apply inhomogenous mag. field
  • Force on particles is:
  • 2 groups of atoms deflected in opposite direction => experience different forces
  • Force must be due to some ang. mom. whose direction has 2 states => spin
56
Q

What feature of compton scattering disagrees with classical?

A
  • Wavelength shift.
57
Q

For non-compatible operators P and Q, if P is first measured giving its eigenvalue p, then Q is measured giving q, if P is measured again, what are the results of measurement?

A

You may yield p again, but this is not certain as P maybe in any of its eigenstates, with some associated probability.

58
Q

What is meant by tunneling?

A
  • In classical physics, a particle with energy less than the barrier potential cannot cross the barrier.
  • In QM, there is a finite probability that the particle may cross this barrier, “tunnel”.
  • Arises from continuity of wavefunction at the boundary.
59
Q

Why does harmonic potentials energy eigenvalues have x dependence?

A

Since potential increases with x

60
Q
A
61
Q

What are the origins of n,l and m?

A
  • n - quantisation of energy
  • m - φ returning to its original value after 2π
  • l - need for legendre polynomial to converge
62
Q

Show that the components of angular momentum do not commute.

A
63
Q

Postulate 1

A
  • Exists wavefn which is continuous, single valued, square-integrable, normalisable.
  • Depends on all system param.
  • Gives all possible predictions of system properties
    *
64
Q

Postulate 2

A
  • Each observable has an associated linear hermitian operator
  • Operators are related by classical eqns.
  • operator has real eigenvalues and orthogonal eigenfunctions
  • immediately fater measurement, wavefn collapses to eigenstate corresponding to eigenvalue.
65
Q

Postulate 3

A
  • Operators are related to each other in the same way as in classical mechanics.
  • Staring point are R and P operators.
66
Q

Postulate 4

A
  • Wavefn can be expanded as a linear combination of its eigenfunctions.
  • This is because they are orthonormal and form a complete basis
  • The mod. square of the coefficients of the eignefunctions give the probability with which each eigenstate is occupied.
67
Q

Postulate 5

A
  • Between measurements of the system, the time-evolution of the wavefn is governed by TDSE.

Physics Completed (8 decks)

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