Advanced Quantum Mechanics Flashcards
What is a condition on the potential of a hamiltonian for it to be translationally invariant?
What about for a system of two particles?
The potential must be constant in r (otherwise the hamiltonian would not be translationally invariant).
-> i.e: there can be no external forces for translational invariance
However, for a system of two (or more) particles, the potential can be a function of the difference of particle locations, so invariant to spatial translation.
Why is the hamiltonian by definition invariant to time-translation?
The hamiltonian is evaluated for an instant in time, and is not dependent on the past or the future.
What is meant by galilean invariance?
The combination of space/time translation and rotational symmetry groups. (all space-time continuous symmetries)
What is the significance of two operators that commute?
They can be simultaneously diagonalised, i.e: the operators SHARE EIGENSTATES.
-> a state can be an eigenstate of both operators simultaneously, so the result of both “measurements” can be known simultaneously.
What is the effect of parity and time reversal transforms on momentum?
Both result in p -> -p (but for different reasons)
What is the form of the lorenz boost operator?
4x4 matrix acting on space-time 4-vector.
y, y, 1, 1 on the leading diagonal
-y beta, -y beta on the top left anti-diagonal
Given transformation operator T, what is the form of the transformation of operator A?
What is a condition for the invariance of operator A under the transformation?
wavefunction –> T wavefunction
! A –> T A T^-1 !
This can be shown by considering the action of A on a wavefunction, and then transforming this.
SHOW (check notes)
For invariance: A = T A T^-1 = T^-1 A T
(the last bit can be shown by considering a matrix element involving A)
What is the difference between a “passive” and “active” transformation?
Passive is the action of an operator on a wavefunction to transform it.
Active is changing the coordinate system of a wavefunction to transform it.
How can we find symmetry generators for continuous symmetries?
Consider the active transformation, making the transformation parameter arbitrarily small (as it is continuous) and making a taylor expansion.
What is the generator for spatial translation?
Show this derivation.
Momentum operator is the generator for spatial translation. (this makes sense: if we add momentum to a system we expect it to translate as time increases)
Check notes.
How can we show that for a hamiltonian to be invariant to some transformation, it must commute with the generator of the symmetry?
Set that the matrix element[acting on transformed wavefunctions] = matrix element[acting on untransformed wavefunctions]
Then make the expansion to first order with the symmetry generator (multiplied by small parameter).
Then set that the terms in the small parameter must sum to 0 (not including the term in the small parameter squared). This will take the form of a commutator being equal to 0.
What is the generator for time translation?
The hamiltonian is the generator for time translation.
Thus the hamiltonian must be time-translation invariant.
What is the form of the hamiltonian operator?
momentum operator^2 / 2m + potential.
also i hbar d/dt
What is the form of the momentum operator?
-i hbar d/dx
or del in multiple dimensions
What is the generator of rotations?
How can we derive this?
Angular momentum operator (in the direction of the rotation axis).
Consider a small angle rotation (around some axis), and it’s action on the x, y and z coordinates. This action can be expressed as a matrix, take the taylor expansion (first order).
^ this expresses the effect on the coordinate axes, to show the effect on a wavefunction, expand the transformed wavefunction to first order also.
This will result in something that can be identified as a cross product, and then put into the form of an:
angular momentum operator (in the axis of the rotation).
(Check Notes)
What kind of transformations do not have generators?
Discrete symmetries to not have generators.
The discrete symmetry operators are defined by their action on the wavefunction.
What are the possible eigenvalues of the parity and time-reversal operators?
What is special about the eigenstates of the time-reversal operator?
Eigenvalue +-1 (as the operators squared give the identity)
Time-reversal eigenstates are either purely real or imaginary. (this can be shown by considering the hamiltonian (energy) acting on:
- the normal state (but then take the conjugate of this equation)
- the transformed state
we can then show that for time-reversal eigenstates, the conjugate state is equal to +- the state.
How can we find the generator for lorenz boosts?
Consider the lorenz boost operator (4x4 matrix) with small delta beta => so y ~= 1.
What are the properties of unitary operators? (ADD TO THIS)
Which symmetry operators are unitary, how can we show this?
Unitary operators preserve the inner product (i.e: the inner product of two states acted on by the operator is equal to the inner product of the un-operated states).
thus the hermitian conjugate of a unitary operator is equal to its inverse.
Symmetry transformations that keep the volume element of space invariant are unitary.
This can be shown by considering the inner product of two (actively) transformed wavefunctions, and changing variables to the transformed basis.
What is an expression for e^x, very useful for finding finite transformation operators?
e^x = lim{N -> inf} [1 + x/N]^N
this can be shown by taking the taylor expansion of either side.
How can we derive finite tranformation operators from infinitesimal transformations?
Consider the finite transformation as an infinite series of infinitesimal transformations. (i.e: lim{N->inf} of the finite transformation parameter/N).
This can be expressed as an exponential.
The result is just e^(the infinitesimal transformation, with the finite transformation parameter instead of the infinitesimal one)
Derive the form of the finite time translation operator.
Check notes.
e^(-i/hbar t H)
What is the commutation rule for angular momentum operators L_i and L_j ?
i hbar levi-cevita{ijk} L_k
Why must we use euler angles to perform a general finite rotational transformation?
We cannot apply the three angle transformations one after the other as the three angular momentum operators do not commute.
If we work in the mindset of performing three rotations one by one from the outset, we do not have this problem. However, we must now consider that the coordinate axes will change after each transformation.
What are the three elementary euler angle transformations?
How can we express the three transformations using operators in the original basis?
Rotate by ALPHA about Z^
Rotate by BETA about X’^
Rotate by GAMMA about Z’^
The second two transformations will involve angular momentum operators in the new bases, which is not ideal:
Check notes.
Rotate by GAMMA about Z^
Rotate by BETA about X^
Rotate by ALPHA about Z^
What is a Wigner D matrix?
How can we construct it?
A Wigner D matrix is the matrix formed of the matrix elements of the euler finite angle operator in the l, m_l basis (use l, m_l in the L_z basis).
We can use this to perform a finite rotation on an eigenstate of L_z.
Note that the action of the rotations about Z^ are simply that of eigenvectors.
Also note that the rotation about X^ will not mix l and l’ states, only m_l and m_l’, so there will be a dirac delta ll’ dependence (not part of the Wigner D matrix).
Check notes.
When finding a Wigner D matrix, how can we find the X^ rotation operator in the L_z basis?
Make a taylor expansion of the exponential, then notice that the expansion can be simplified into an initial term and two trig terms ( x matrices).
What is the commutator [r, p]?
= i hbar
for r and p being parallel
How can a cross product be expressed in einstein summation notation?
c = a x b
c_k = levi cevita {ijk} a_i b_j
How can we represent spin-1/2 matrices in terms of pauli matrices?
S = ( hbar / 2 ) pauli matrices
What is the expression for general pauli matrix multiplication?
pauli_i pauli_j = delta{ij} I + i levi-cevita{ijk} pauli_k
i.e: pauli^2 = I
pauli[i] pauli[j] = -pauli[j] pauli[i]
What is the result of ( pauli dot momentum operator )^2 ?
What about (pauli dot O)^2 where O is a general operator?
simply p^2 I
(as the condition p x p = 0 is satisfied)
(pauli dot O)^2
= O^2 + i pauli dot (O x O)
Do the pauli matrices commute with the position and momentum operators?
Yes
What can e^(i d [pauli_j]) be evaluated to?
cos(d) I + i [pauli_j] sin(d)
