TQW Pt. II Flashcards
(145 cards)
1
Q
Free particle wavevfunction calculation.
A
Free particle properties. Note that you can get rid of one of the wavefunction terms (in this case B) since the +ve or -ve just represents the direction of motion.
2
Q
A
3
Q
A
4
Q
By ‘what is’ I mean how can you express?
What is position in 2D systems?
What is momentum in 2D?
What is del in 2D?
A
5
Q
True or false, you can seperate the vector momentum operator into N components (where N is the number of dimensions)?
A
Breaking the momentum operator into its components can make N-dimensional problems a lot easier, rather than doing it all in a vector. I.e. with the free particle in 2D.
6
Q
What is the relation between the Hamiltonian and Momentum operator (when Potential, V= 0)?
A
7
Q
What is the commutation between the Hamiltonian and Momentum operator for a free particle(V=0)? Answer for each of the components of the Momentum Operator.
A
8
Q
T/F? The eigenstates/eigenfunction of the Hamiltonian and Momentum operators are the same. Why is this the case?
A
True, the eigenstates of the Hamiltonian and Momentum operators are identical. This is because the commutation of the Hamiltonian of a free particle and Momentum are compatible (commute to 0).
9
Q
Starting with the equation of a plane wave (in 2D), show that it is the eigenstate to the Momentum operator and find the eigenvalues of momentum.
Do the same with the Hamiltonian- remember the del^2 operator is not a vector.
A
Vector momentum tip, break the vector momentum operator into its components, Px[hat] and Py[hat], allow both of these to act on the wavefunction, not forgetting to state their direction/unit vector when stating the final answer.
Note: Error in slide; corresponding momentum eigenvalue should be
hbarK = hbarKxex + hbar*kyey
10
Q
What are the boundary conditions for a particle in a 2D infinite potential box of units of length Lx * Ly?
What is Schrodinger’s equation for this wavefunction?
A
11
Q
If you are seeking a seperable solution, how can you reduce (10) to 2 independent 1-Dimesional equations?
A
Since you are effectively applying the hamiltonian to each dimension individually, they will return eigenvalues epsi(x) and epsi(y), which is NOT the energy eigenvalue of the eigenstate, only a fraction of it.
The energy eigenvalue = sum of eigenvalue for each dimension i.e. E = epsi(x) + epsi(y).
12
Q
A
13
Q
A
Hints:
Born Rule Pn = I Cn I^2
Total probablility = 1 = P(3,2)+P(3,2)+P(4,3)
- Find what L is equal to
- Use this to find the probabiliy of each state (apply into born rule equation again).
- Expected energy = sum of (Pn*E(n,m))
14
Q
What is the wavefunction of a 2D box?
A
15
Q
What are the energy eigenstates in a 2D box?
(The equation of)
A
16
Q
What does it mean for states to be degenerate?
What does this mean for the superposition of 2 Eigenstates?
A
Degenerate states have the same Energy eigenvalue.
Superposition of degenerate eigenstates is also a eigenstate.
17
Q
How can you lift the degeneracy of a system?
A
Breaking symmetry lifts degeneracy, i.e. changing dimensions of container.
18
Q
If Ia> and Ib> return the same energy eigenvalue, is their sum also an eigenstate?
A
If Ia> and Ib> return the same eigenvalue then they are degenerate. Any combination of them will produce an eigenstate.
19
Q
State an orthogonal combination of
(u4,3 - u3,4)/sqrt(2).
A
20
Q
Are degenerate eigenstates orthogonal to each other eigenstates of the same degeneracy?
What about to eigenstates of a different degeneracy or that return different eigenvalues?
A
Eigenstates of the same degeneracy may or may not be orthogonal since they do not return different eigenvalues.
Degenerate eigenstates are always orthogonal to all other states.
21
Q
Suppose the particle is in an arbitrary state Iψ>, what is the probability that a measurement of its energy will give a particular result?
(a) that the measurement is not an eigenvalue of the Hamiltonian.
(b) that the measurement is an eigenvalue.
A
22
Q
For a square box:
What combination of states yield 2ϵ , 5ϵ and 65ϵ . For each, what would the probability be for yielding each of the results if the particle is in an arbitrary state?
Hint for 65ϵ (4,7) and (1,8) both have a sum of squares of 65.
A
Don’t forget potential degeneracies!!
23
Q
After the measurement of an arbitrary wavefunction, what happens?
What happens if the wavefunction it collapses into is degenerate i.e. give the wavefunction after the collapse in dirac-delta notation.
Do this for the eigenvalue of 5e.
State the normalisation constant of this state.
A
After measurement, wavefuction collapses to reflect the measurement obtained.
24
Q
Give the form of a central force in terms of a central potential. Hint: Vector calculus notation.
A
25
True or false? A particle subject to a central potential experiences no torque?
DON'T LOOK AT IMAGE UNTIL YOU ARE READY FOR ANSWER.
True!
26
State the equation for a harmonic potential for a Simple harmonic oscillator in terms of r, then state in cartesian coordinates.
27
What is the Hamiltonian for a 2D SHO?
28
What is the Shrodinger equation for the SHO in terms of vector r and cartesian coordinates?
29
What is the energy eigenvalue equation for a SHO in 1D and 2D?
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31
State the Cartesian Hamiltionian to the SHO in terms of Momentum and Position operators.
State the SHO TISE in 2D SHM.
32
The lowest state of a SHO.
33
What is the angular momentum operator, Lz, in cartesian coords?
Is it commutable with the Hamiltonian?
This means that the Hamiltonian and this operator share the same eigenstates. However, the degenerate energy eigenstates which all have the same energy eigenvalue may correspond to different eigenvalues of the commuting operator
34
What is the angular momentum operator, Lz, in polar coords?
Is it commutable with the Hamiltonian?
35
What is the TISE for psi(r, phi)? -- TISE in Polar coords for QHO.
Re-write this in terms of the angular momentum operator.
36
What is the general form for an eigenfunction of the hamiltionian in terms of r and phi?
What is the effective potential of the radial shrodinger equation? Hint: writing out the schrodinger equation
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38
Read-
What is the form of an eigenstate of the Angular momentum operator?
True or false? Individually, the first two degenerate eigenstates are NOT eigenstates of the Angular momentum operator, but a combination of them is?
True! 1/sqrt(2) * (u1,0 +/- i*u0,1) which is of the form of the eigenstate of the angular momentum operator
phi(phi) = 1/sqrt(2pi) *e^(i*m*phi)
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40
Using the wavefunction of a SHO, and the Schrodinger equation in polar coordinates, find a form of the effective radial Schrodinger equation and the effective potential.
To go from eq(4) to eq(5), sub in eq(6) then divide by PHI(phi).
41
What combination of these degenerate energy eigenfunctions will produce an eigenfunction of both the Hamiltonian and angular momentum operators?
42
What is the degeneracy of a SHO?
43
What is the Hamitonian for a Particle moving in a RING? Write in terms of the angular momentum operator.
Hint: Use the moment of inertia of a particle travelling in a circle.
What are the Possible energy Eigenvalues?
Hint: Use the generic form for the eigenfunction of the angular momentum operator.
No radial motion (dPhi/dr = 0, only 1 angular component, phi). Only 1 angular component phi because limited to orbit in one plane, i.e. d(theta)= 0 as well.
44
What are the Lx, Ly and Lz operators in terms of position and Linear momentum operators? (think cross product)
What are the results of the image?
What does this mean about the eigenstates of Lx, Ly and Lz collectively?
Lx, Ly and Lz do NOT share the same eigenstates collectively, due to being non-commutable.
45
What is the square magnitude of the angular momentum operator Lhat ? (Hint: dot product)
What is the result of the square magnitude of angular momentum commutated with Lx, Ly and Lz?
What is an important consequence/property that arises due to this commutation?
Does this mean Lx, Ly and Lz share the same eigenstate?
This tells you that the L^2 operator shares the SAME eigenstates as the Lx, Ly and Lz operators (due to commutation). This means that L^2 and any one of Lx, Ly and Lz can be observed (typically choose Lz).
NO, just because the L^2 operator is commutable with every component operator, we know these component operators are non-commutable. This means we can obsertve the L^2 and ONE of the component operators at any time.
46
What is an important consequence/property that arises due to this commutation?
Does this mean Lx, Ly and Lz share the same eigenstate?
This tells you that the L^2 operator shares the SAME eigenstates as the Lx, Ly and Lz operators (due to commutation). This means that L^2 and any one of Lx, Ly and Lz can be observed (typically choose Lz).
NO, just because the L^2 operator is commutable with every component operator, we know these component operators are non-commutable. This means we can obsertve the L^2 and ONE of the component operators at any time.
47
What are the ANGULAR MOMENTUM raising and lowering operators?
What do they do?
Write a general function in the form
u(r, theta, phi) where u is an eigenfunction of Lz. (Operator LHS, Result/eigenvalue on RHS of eq).
m = 0, +/- 1, +/- 2...
48
Proof the first line.
Next:
Show the result of applying either of the raising operators on some eigenstate, u, of Lz, followed by Lz operating on it.
What do these results show we have found?
The results show that we have found a quantum ladder of Lz eigenstates, each spaced by units of hbar.
49
Before looking at image, what is [L^2, L+]?
What does this mean?
What is L^2[L+[u]]?
Now looking at previous answers in image:
What do the results tell you? Hint: Which eigenvalues change, and which do not?
So the raising and lowering ladder operators change only the eigenvalue of the z-component
operator Lz but not the eigenvalues of the angular momentum magnitude squared operator L^2.
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51
What is the effect of applying the angular momentum lowering operator on the state with the lowest quantum number l'? Raising operator on the state with highest quantum number, l?
State the eigenvalue eq of the L^2 operator.
State the eigenvalue eq for the Lhat operator.
The last line provides important information for our analysis. It say that if the angular
momentum operator L acts twice on a particular state u, we obtain the same state again times a scalar value l(l+1)* hbar^2, so this is an eigenvalue - eigenstate equation for the angular momentum operator L. Hence acting once must yield sqrt(l(l+1)) *hbar u.
52
53
54
For the eigenvalues of the z-component of angular momentum, what range of values can m take?
What is l?
m can take on values between -l (sometimes written l') and l in interger steps, where l is the maximum interger numbr m can take. Applying the raising operator onto the state u(m=l) = 0 = lowering operator on u(m= -l) = 0.
l is the orbital quantum number.
55
Do all Lz on u_m have indentical L eigenvalues independent of the choice of m?
What is teh degeneracy for Lz eigenstates?
Yes! All Lz u_m have indentical eigenvalues independent of the choice of m.
Lz degeneracy = 2l+1 eigenstates, always odd number.
56
READ- Then answer
What is meant by l?
What are the greatest eigenvalues of the Lz operator?
What is the degeneracy of the Lz operator?
What is the same eigenvalue for the momentum operator L?
What does this eigenvalue represent?
Note: What are the greatest eigenvalues of the Lz operator? is -l hbar to l hbar, NOT just -l to l.
What is the uncertainty of the Lx and Ly operators?
57
Using this find the length fo the L hat operator.
58
59
What are the (5) steps to finding the eigenfunctions of L^2 and Lz operators? (Y l,m)
Ang. Momentum III lecture
1. Let Y l,m(theta, phi) =
Theta l, m(theta) * Phi m(phi) (seperable).
2. Apply Lz operator to Y l,m to find
Phi m(phi).
3. Apply L+ operator to Y l, l (this = 0 since l is highest state) to find theta l, l.
4. Find normalisation constant by normalisation (integration of [mod(Y l, l)]^2 = 1)
5. Use the lowering operator on Y l, l to get to the desired momentum eigenfunction +
re-arange.
60
Find
Hint: card 57
Lecture: Angular Momentum III
61
62
Which operators are NOT hermitian? (Are not complex conjugate of themselves?)
LADDER OPERATORS!!! or any operator involving complex values.
63
What is the expectation value for the L^2 operator?
IGNORE SQUARE ROOT
64
REMEBER LADDER OPERATORS ARE NON-HERMITIAN. O =/ O cross
65
What are the eigenvalues for the lowering and raising operators L- and L+?
66
What is the coefficient c l, m for some wavefunction Phi(theta, phi)?
67
Before looking at image, what is the complex conjugate of a spherical harmonic?
What are the Orthonormality conditions for spherical harmonics? (State Dirac notation and integral notation- Should be the same result as always)
Hint:
If Normalised and non orthonormal, overlap = 1, if orthonormal, overlap = 0.
Do the Hamiltonian and Squared angular momentum operators commute?
Does the potential and angular momentum operator commute?
Does [H, L^2] = 0?
Does [H, Lz] = 0?
Y, Yes both commutation relations are true, meaning H, L^2 and H and Lz operators share the same eigenfunctions.
68
Given this is the angular momentum operator L in spherical coordinates, (not in a specific direction), what is the operator L^2?
Don't forget, the Angular Momentum operator is the cross between operators R and P (Radial distance and Momentum).
69
What is the Angular momentum operator L in spherical coords?
Don't forget the -i hbar from the momentum operator!
Also don't forget P = -i hbar [DEL OPERATOR]
^ Where the differentials come from.
70
What is the shroedinger equation for the motion of a particle on the shell of a sphere? Hence, what is the hamiltonian acting on Psi for this case?
(Ignore potential)
Re-arange to find the Eigenvalue in terms of the moment of inertia.
Hamiltonian is the same thing as the second line of equations/Shroedinger equation for spherical shell. Hamiltonian on wavefunction = TISE
71
What are the possible energy eigenvalues for a particle travelling on a spherical shell?
Requires substitution into hamitonian of particle on a spherical shell.
Also find the difference in energy between succesive energy levels?
72
What is the Hamiltonian for the rigid rotor?
Hint: Centre of mass and relative component.
H = Hrel + Hcm
Hcm = Pcm/ 2M
Hrel = prel/2mu + V(r)
73
Use separation of variables for the 3D shroedinger equation in spherical coordinates to produce 2 equations, one being radial and the other being angular.
A = l * (l + 1)
74
We have already solved the angular equation, but what about the radial one?
The radial component eigenfunctions are bessel functions.
75
Given the information for the rotational spectrum for Carbon Monoxide and the Hamiltonian acting on a free particle (assuming r is constant), find the bond length between 2 atoms
A photon of frequency 1.16 *10^11 excites the molecule from l=0 to l= 1.
Hint: Quantum rigid motor.
76
There is a 50% chance for any measurement to be from the I 2, 1> state and 50% chance for the I 2, -1> state. Since both these states return the same eigenvalues for the Hamiltonian and Angular momentum squared operators, these eigenvalues have a (50 + 50) 100% chance of being measured. The Lz Eigenvalues are different therefore there is a 50% chance for - hbar and 50% for + hbar measurements of Lz.
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For a diatomic molecule, assume d/dr = 0, V(r) = 0.
As d/dr = 0, V(r) = 0 , H Phi = L^2 Phi --> l(l+1)hbar^2/ 2 mu r^2 = E
80
To find an eigenstate you will have to match it to its known Bessel functions.
81
To find an eigenstate you will have to match it to its known Bessel functions.
To find an eigenstate you will have to match it to its known Bessel functions.
82
What is the force a particle trapped in a spherical potential well exerts on the walls? The wavefuction is confined to a sphere of radius r0 (V(r)= 0, d.dr = 0).
83
State the eigenenergies inside an spherical symmetric potential.
*Analogous to Infinite potential box.
State boundary conditions in terms of a Bessel function J_l.
State for lowest order given J0(r) = sin(r)/r
Read after (following and image)
For the lowest order Bessel function, zeros can be found at x = n*pi, n=1,2,3 (not 0 due to 0/0).
* when l = 0, k = (pi)*n_r / r_0 ....
r_0 is the radius/boundary, analogous to square well of length L.
DONT FORGET THAT j_l(K r) --> Bessel for spherical potential well has input k*r.
k = n*pi/r0 !!!!! ---> ALWAYS sub into radial spherical potential questions.
84
What is the overlap integral of
I Y(l = 0, m = 0 ) I**2?
Y(0,0) = C* (sin theta)^0 * e^(i * 0 * phi) = 1
85
* Within sphere of radius (1/2)*r0.
Since un-normalised - need to find the result integrating from 0 to r0, and from 0 to r0 /2. Then to find Prob for r
86
What is the Hamiltonian of the Hydrogen atom (in terms of momentum operators)?
Write as 2 time-independent Schrodinger equations.
87
State the radial equation of the HYDROGEN ATOM.
(The equation derived from Shrodinger equation and applying seperation of variables).
DON'T forget the COULOMB POTENTIAL!!!
Applying the angular equation to Y(theta, phi) allows Y to be cancelled out.
88
What is the effective potn for the Hydrogen atom?
Summary of Image on next page as well.
-As r--> inf, Veff(r) -->0. If electron has positive energy it is unbound.
- Unbound states are called scattering states.
- All bound state electrons must have negative energy eigenvalues.
- Boundary between positive and negative eigenstates is called the ionisation energy.
- Classically forrbidden region is where KE is negative.
- In ground state, l=0, re-arranging effective potential and setting equal to the KE tells you the classical turning point -->
r > e**2 / 4*pi*epsilon*Energy
^more detail slide 89.
- Since centrifugal potential increases with l, we expect effectice potential energy that determines radial motion to be different for each state with different angular momentum, l.
89
90
DO NOT NEED TO REMEMBER
DO NOT NEED TO REMEMBER
91
State the the relation between jmax, l and the Principal quantum number.
State the recurrence relation between the coefficients of the radial eigenfunctions of the Hamiltonian.
92
What is the truncated radial polynomial sum for the radial equation of the Hydrogen atom?
What is l, jmax and n for the hydrogen groundstate?
*Magnetic quantum number does not effect radial equation, only orbital quantum number l since it effects effective potential.
* Energy depends only on Principal quantum number n-> Rydberg equation.
What is the full radial equation for a hydrogen atom in its ground state?
93
What is the equation for the energy levels of a hydrogen atom?
What is the Rydberg energy constant equal to?
94
What is the radial wavefunction for the 1st excited state with principal quantum number n = 2 for a Hydrogen atom.
95
What is the result of applying the Hamiltonian onto the Hydrogen eigenfunction u(r, theta, phi)? Result for L^2 and Lz operators?
[Look at image]
What is the normalisation condition for the complete Hydrogen wavefunction?
[Give in Dirac and Integral form]
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100
What is the general solution to the radial equation of the Hydrogen atom?
101
When are radial equations for a Hydrogen atom orthogonal, and when are the angular components orthogonal?
If checking for the orthogonality between two hydrogen eigenstates- which is the easiest/best component to test orthogonality for?
It is best to check orthogonality for the ANGULAR equations, Y(theta, phi) since they are always orthogonal for all l and m!!!
Where as R(r) is only orthogonal for same l.
102
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104
105
How do the eigenstates and eigenvalues change for the hamiltionian in a magnetic field, B?
What is the equation for the energy eigenvalue in a magnetic field for the Hydrogen atom?
Energy eigenvalue includes a magnetic moment term!!
Eigenfunctions remain the same since H is commutable with Lz.
106
What is spin?
What are the eigenvalues of the S^2 operator (and hence S operator)?
What are the eigenvalues of the Sz operator?
107
What is the Sz eigenvalue of an electron?
108
Does Spin depend of spatial variables?
What is the spin of an electron?
What are the eigenvalues of the Sz operator for an electron?
How do we write the 2 corresponding eigenstates for the electron?
109
110
How do we represent a combination of - and + spin quantum states?
What are the corresponding vectors for - and +?
111
How can we write a general spin state in terms of (first) a and b, and (second) Psi?
<+ I Psi> = a, <- I Psi> = b
112
What is the answer?
113
What is the probability of measuring the spin - or spin + state? Given Psi = a I-> + bI+>
P(+) = a^2, P(-) = b^2
114
n
115
What is the probability of measuring either - or +hbar/2 if Sz has already been measured to be hbar/2?
116
Sz and Sx are non-commutable so when measuring Sx after Sz you remove all previous information on the previous eigenstate and measure a completely different eigenstate.
117
From these measurements determine modulus squared of a,b,c and d.
Are I-/+>x just superpositions of I-/+>?
Re-write I->x and I+>x with relative phases alpha and beta.
Yes I->x and I+>x are just superpositions of I-> and I+>.
118
Re-write I->x and I+>x with relative phases alpha and beta. Find the relationship between alpha and beta. Choosing an appropriate value for alpham state the final form of I->x and I+>x in terms of I-> and I+>.
119
What is the matrix representation of Sz? What is the matrix representation of Sx?
Do Sz, Sx, and Sy share the same eigenvalues?
What are the possible eigenvalues?
Yes Sz, Sx and Sy all have eigenvalues of +/- hbar/2.
120
121
What is are the eigenvalues of the S- and S+ operators?
What do they reduce to for spin 1/2?
Write them in matrix form.
122
123
124
Hint: Consider for both I-> and I+>.
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126
The beam leaving the SG apparatus is in a pure I+>x state which is a superposition of I+> + I-> with equal probabilities--> beam splits into two beams of equal intensity.
127
128
What is the eigenvalue of S^2 operator? What is it equal to in Matrix form?
Eigenvalue = 2hbar^2 /4, Matrix form 2*hbar^2 /4 * I^.
129
What matrix describes some spin in some random direction n? Express n as a vector in 3D polar coordinates?
130
Example 8.5: A beam of spin- 1
2
particles is sent through a series of three SternGerlach apparatus, as shown in Fig. (8.12) . The second Stern-Gerlach apparatus
is algned along the ~n direction, which makes an angle θ in the x − z plane with
respect to the z−axis.
1. Find the probability that particles transmitted through the first Stern-Gerlach
apparatus are measured to have spin down at the third Stern-Gerlach apparatus.
2. How must the angle θ of the second Stern-Gerlach apparatus be oriented so
as to maximise the probability that particles are measured to have spin down
at the third Stern-Gerlach apparatus. What is this maximum fraction?
3. What is the probability that particles have spin down at the third SternGerlach apparatus if the second Stern-Gerlach apparatus is removed from the
experiment?
1. P(+n projected on +z)*P(+z projected on +n)
131
What are the eigenstates spin 1 particles can take? How do we represent these states in bra-ket notation? In vector notation?
132
133
Only eigenstates of the Hamiltonian are states. Any operator with the EXACT same eigenstates as the Hamiltonian will also produce stationary states.
134
What is the time independent Schrodinger equation?
135
By using the Time independet and dependent shrodinger equations, write cn(t) in terms of cn(0). Hence state the full wavefunction as a product of cn(0), time evolution and Iun>.
** Multiplying by summation.
136
State the Spin Dynamics Hamiltonian in terms of a dot product, then in a matrix representation. What are the ENERGY eigenvalues of the I-> and I+> states?
Pay attention to the +/- signs! E+ is negative, E- is positive.
137
Provided this state, determine if the total state is time dependent.
138
Show if the phase has a measurable effect, and if it is an eigenstate.
139
For this state, find the probability to measure the spin component along the positive z-axis.
140
For this state find P( I+>x )
141
Calculate the expectation value for Sz for this state.
142
State the Ehrefrest theorem for an expectation value!
143
Use the Ehrenfrest theorem to calculate the time dependence of expectation values of the three spin components.
144
How can this be written more compactly? What are the general solutions?
145
Given the probabilities for each corresponding eigenvalue, find the expectation value .
