Problems Flashcards
(159 cards)
1
Q
Dirac for Hermitian
A
Ket only
2
Q
For operator A, what does A^2 mean?
A
- Apply A twice
3
Q
How to use given eigenvalue equations with operators + to solve
A
- Create equations in terms of operator and λ only
E.e A^2 = 2 *identity matrix
- A Ψ = λ Ψ
- A^2 Ψ = AA Ψ
3 Sub 1 into 2
- A^2 Ψ = A λ Ψ = λA Ψ
- Sub in 1. Again
- A^2 Ψ = λ^2 Ψ
4
Q
How to prove position and momentum commutation
A
- Write out commutation relation
- Find x on px i.e x operator on -i ℏd Ψ/dx
- Find p on x i.e. -i ℏd/dx on x Ψ
5
Q
Pauli Spin x
A
ℏ/2 *
( 0 1)
(1 0)
6
Q
Pauli Y matrix
A
ℏ/2 *
( 0 -i )
( i 0 )
7
Q
Pauli Z matrix
A
ℏ/2 *
( 1 0
0 -1 )
8
Q
How to calculate probability that a measurement of e.g. x component of spin gives set value
A
- Express spin state in terms of Sx eigenstates ->
- Square the function ( NB if only +Ve value, only need to look at spin up etc)
9
Q
How to express spin state in terms of S eigenstates
A
- Write out eigenstates for up and down separately e.g. (1,0) and (0,1)
- Form ansatz of: |Ψ> = α (1,0) + β (0,1)
- Solve matrix: |Ψ> = (α, β)
- Compare with spin state to find values for α and β
- Sub in values for α and β from 4 into equation 2
10
Q
Eigenstates of Sz
A
I+> = (1,0)
I-> = (0,1)
11
Q
Eigenstates of Sx
A
I+x> = 1/sqrt2 (1,1)
I-x> = 1/sqrt 2 (1,-1)
12
Q
Eigenstates of Sy
A
I+y> = 1/sqrt2 (1, i )
I-y> = 1/sqrt 2 (1,-i)
13
Q
Eigenvalue matrix
A
AI - λI = 0
AKA
A Ψ - λ Ψ = 0
14
Q
Where does λ go for 3x3 eigenvalue problem
A
diagonal upper left to bottom right
15
Q
How to calculate potential energy values of Hamiltonian
A
Find enigenvalues
16
Q
Calculate normalised eigenvectors
A
- Find eigenvalues
- Find eigenvectors
- Normalise eigenvectors (divide by magnitude - squareroot of sum of squares)
- Express 3 with 4
17
Q
Determinant of 3x3 matrix with 2 zero elements in first row
E.g:
- 0
- 2
- 1
A
Det = (5- λ) * [1 - λ. 2. ]
[ 2. 1- λ]
- Find determinant of 2x2 matrix and multiply by (5-λ)
18
Q
How to calculate uncertainty in operator?
A
ΔP = SQRT of ( <P^2> - <P>^2)
- find expectations with Born’s rule
19
Q
For which model does n=0 for the ground state
A
Harmonic oscillator
20
Q
How to find eigenvectors
A
- Find eigenvalues
- Sub back into matrix A
3 Multiply new matrix A by e.g. x,y,z to form simultaneous equations e.g. 3 for 3 by 3 matrix
- Use solutions for values for x,y,z to form new vectors (magnitude only) e.g if x= 0 and y=z -> 0,1,1
21
Q
Find probabilities of measuring eigenvalues from Hamiltonian matrix
A
- Normalise given wavefunction
- Calculate Cn I.E. Find inner product of each eigenstate with normalised wavefunction
- Probability (λ) = |Cn^2|
22
Q
Find expectation value of Hamiltonian matrix if only 1 eigenvector
A
- Find H |Ψ> I.E. multiply Hamiltonian matrix and normalised eigenvector
- Find <Ψ| H |Ψ> I.E. a) Find complex conjugate of normalised eigenvector b) multiply this by H |Ψ> (1. answer)
23
Q
Find expectation value of Hamiltonian matrix if more than 1 eigenvector
A
- If eigenvectors form an orthonormalisation basis - state is expressed as linear combination
<H> = lC1l^2 E1 + lC2l^2 E2 + l C3l ^2 E3
- Verify inner product = 0
2. Square coefficients in front of states of wavefunctions CN - can also be found from inner product of normalised wavefunction and each state
3. Sum of CNs multiplied by each value for energy = expectation value
</H>
24
Q
Why only consider electron for Stern Gerlach?
A
Magnetic moment proportional to Q/2m x angular momentum operator
- As electron is lightest, it dominates
25
Why gradient of B field needed for Stern-Gerlach?
Force = - ∇Potential energy = -∇ (mu.B)
No force in electron with no gradient
26
Expectation value of a matrix
E.g of y spin
o
oR
if have a linear combination of eigenfunctions, square and sum prefactors
27
Components of angular momentum - cyclical permutations used for proof of commutation
-Lx = - iℏ (y d/dz - z d/dy)
- Ly = -iℏ (z d/dx - x d/dz)
- Lz = -iℏ (x d/dy- yd/dx)
28
Eigenvalue of L^2
L^2 = ℏ^2 l (l+!)
29
How to prove L^2 commutation
- Use [A^2, B] = A[A,B] + [A,B] A
30
Lz eigenvalue
ℏm where m = -l to l
31
What is εijk
- Levi-Civita symbol - totally anti symmetric tensor
32
Position momentum commutation relation
[Xi, Pj] = i ℏ δij = - [Pj , Xi]
33
Commutator of orthogonal position
[Xi , Xj ] = 0
34
Commutator of orthogonal momentum
[ Pi, Pj ] = 0
35
2 important commutation equations
[ A, BC ] = [ A, B ]C + B [A,C]
[ AB, C ] = A [ B, C] + [A, C] B
36
How to prove commutation relation for angular momentum
- Using identities in form of Lz = xPy -yPx
- Expand using 2 important commutation relations
- Cancel identities which commute
37
What would expectation value for Hamiltonian be with spin =3/2
< 3/2 ,m sub s l H l 3/2, m sub s>
Where m could be -3/2, -1/2, 1/2, 3/2
1. Apply Hamiltonian and expand/cancel operators where possible as < 3/2 ,m sub s l H 3/2, m sub s>
2. As H is now a constant, move outside: H < 3/2 ,m sub s l 3/2, m sub s>
3. If brakets are orthonormal, =1
4. E then = constant obtained and moved in part 2.
38
Operator forS^2 acting on l s, m sub s>
ℏ^2 s(s+1)
39
How to find if energy levels are degenerate?
- Plug in all possible values for m
- If any repeat -> degenerate
40
Why probability density goes to zero at infinity?
Necessary condition for it to be normalisable
41
Which part of bracket is row vector?
Bra -> wouldn’t want to row without one
42
Which part of bracket is column vector?
Ket - building columns of ketamine
43
How to normalise a state e.g. spin
- Find bra (row vector)
- Inner product
44
Matrix multiplication
(A B ) (E)
(C D) (F)
(AE + FB)
(CE + DF)
45
What is value of Sx^2
ℏ^2 / m
46
How to transpose matrix
Rows become columns, columns become rows
AB = AC
CD BD
47
If matrix only has diagonal components (zero elsewhere), find eigenvalues
The diagonal numbers
48
Cos AB
Cos A Cos B - Sin A Sin B
49
50
Consideration if told w>>>> ∇w
That at some stage should use this to reduce w+ ∇w to w and make approximation through cancelling
51
Exponential part of spherical harmonics Meaning
Azimuthal part
52
M
Magnetic quantum number
Z component of angular momentum
53
Curly l
Orbital quantum number
- Total angular momentum
54
The Legendre Polynomial
Pml cos θ part of spherical harmonic - > depends on θ
55
Meaning of l U > = lA> lB> lC>
A = quantum number, n
B = orbital quantum number, curly l
C = magnetic quantum number, m
56
57
Expectation energy for hydrogen in state expressed as l U > = lA> lB> lC>
- A = n —>
- 13.6/ prefactor squared * (∑ 1/n^2)
58
Inner product in Dirac notation
< Ψ l Ψ > = < AF l AF > = A*A
59
How to find operator matrix given eigenvalue and orthonormal basis?
Matrix multiplication - dummy 2x2 matrix x basis = eigenvalue x basis e.g basis of 0.1
[A B ] [0 ] = λ [0 ]
[C D ] [ 1]. [1 ]
Solve as simultaneous equations
60
Difference between time dependent and time independent Schrödinger
- Time independent: LHS = E (energy) x Ψ
- Time dependent: ih dΨ/dt = E -> becomes LHS
61
Derive arbitrary spinor given normal vector of spinor called Sn
Multiply by Pauli matrices
- ℏ/2 [ Snx σx + Sn σy + Sz σz]
Solve each part individually to get 3 2x2 matrices, then add them
62
Eigenvalue for S^2
- ℏ^2 s ( s +1 )
where s = spin quantum number, 1/2
63
64
65
Features of ISW
- Goes from 0 to L
- Wavefunction = 0 at borders
66
Why must wavefunction be continuous?
- Otherwise second derivative (in Hamiltonian) = singular (discontinuous or non-physical)
67
Important features of finite square well
- Tunnelling - exponential decay outside well
- Length = 2a
- Potential is -Vo inside well and 0 outside
68
What happens to potential in ansatz for k outside FSW
Disappears because V = 0 there
69
Which general solutions outside FSW disappear and why
- Before well -> Ae^-kx
- After well -> Ge^kx
Both would cause wavefunction to blow up and there is an exponential decay outside well
- needs to die off fast enough to keep integral finite
- these solutions would not be normalisable
70
Energy/potential relationships in FSW
- Outside EV
71
E + Vo inside and outside FSW
Difference due to potential
- INSIDE: E + Vo > 0 as E is negative but Vo is larger and positive
- OUTSIDE: E + Vo <0 as E is negative and V=0
72
73
ISW energy eigenvalue
En =(n^2 π^2 h^2 ) / ( 2M L^2)
L = length of box
74
General solution for K in FSW
K = √ ( 2m ( En + Vo ) ) / h^2
75
Why does quadratic term dominate in Taylor expansion of harmonic oscillator?
- X^1 = 0 because 1st derivative is a minimum as d/dr
- X^2 = constant
- Others closer to 0
76
Steps to solve FSW
1.Drawing
2. Ansatz -> E.g. Ae^-aX + Be^ax inside well, outside, Ae^ikx + Be^ikx
3. Check continuity (x=0) for ansatz and their derivatives to form simultaneous equations
4. Solve equations for coefficient of interest as a fraction of magnitudes
5. Square the result
77
Wavefunction ansatz for FSW
E.g. Ae^-aX + Be^ax depending on location
Inside well
Outside:
- Ae^ikx + Be^ikx
78
FSW if E>Vo and transmitted
- k becomes k’ as energy changes
- E - Vo = h^2k^2/2m
- Flux is used (density * velocity) so present answer as e.g lCl^2 solution * k’ / ( lAl^2 solution *k
79
How to verify result for reflected and transmitted flux when solving FSW?
Total of answers = 1
80
How to create equations to find energy of harmonic oscillator
1. H =-h^2/2m d2/dx2 + 1/2 m w^2 x^2
2. Substitution of y^2 = (mw/h) x^2 with rescaled energy (E = 2E/hw)
3. H =-h^2/2m (mw/h) d2/dy2 + 1/2 m w^2 (h/mw) y^2
4. Wavefunction equation (where hw/2 cancels) = E * Ψ
5. -d2 Ψ/dy^2 + (E - y^2) Ψ (y) = 0
81
Solutions to energy of harmonic oscillator
Asymptotic and full
82
Asymptotic solution
To find energy of harmonic oscillator
- Consider behaviour as x-> ±∞
- Ensure as y -> ∞, x -> ∞ —> do so by having E remain small i.e y>>>E
Solution = d2 Ψ(y) /dy^2 = y^2 Ψ(y)
- Ψ has infinity as superscript to denoted asymptotic
83
Ansatz for asymptotic solution
Ψ(y) = Ae^(y^2/2) + Be^-(y^2/2)
- where A part cancels due to boundary conditions
84
Other term for asymptotic solution???????
Finite part of harmonic oscillator function
85
Steps to find position within barrier of FSW when given values for energy and potential and PROBABILITY (e.g. 50%)
1. Write wavefunction ansatz for both positions and relate to probability
= l D e^ - αX2l ^2 / l D e^-αX1 l ^2 = 0.5
Where particles starts at X1 and is at X2 for 50% probability
2. Rearrange for Δx
3. Use α values and given values to solve numerically
86
Alpha in finite SW
α = √ (2m (Vo - E) ) / ℏ^2
87
What does small tunnelling distance signify ?
Very unlikely
88
Which formula for tunnelling probability
= lFl^2 / lAl ^2
89
Consideration if calculating quantum tunnelling probability
If E>>>V or vice versa, cancel terms in fraction before calculating
SI units!
90
What is tunnelling probability also sensitive to ?
V and b (length of well)
91
Calculate time until fusion, assuming regular intervals for collisions
1. Number of collisions = 1/T (quantum tunnelling likelihood)
2. Time until fusion = interval between collisions/ T
92
How to find solution for k in finite square well
1.Ansatz Ae^-ikx
2. Sub into TIWE
3. Rearrange
93
How to correctly compute commutators
Ensure each acts on each other - not just algebra!
94
How to find eigenvalue for operator when given a wavefunction
1. Apply operator to wavefunction
2. Equate 1 to typical eigenvalue statement
E.g for operator H
H |Ψ> = λ |Ψ>
3. Cancel terms, then state λ = (whatever equates to known eigenvalue on other side of equation)
95
How to prove sum of expansion coefficients (Cn) = 1
1. Write normalisation equation(integral)
2. Substitute in Wavefunction expansion (DIFFERENT COEFFICIENTS)
∫ ( ∑ Cn Un )* (∑ Cm Um) dx = 1
3. Bring constants outside
Cn*Cm∫ ( ∑(Sub_n) Un )* (∑(sub_m) Um) dx = 1
4. State orthonormality equation
∑(sub_nm)Cn*Cm δnm = 1
5. Only true when n=m, therefore
∑(sub_n) lCnl^2 = 1
96
How to find principal quantum number given a wavefunction
Apply Hamiltonian - may need radial if angular momentum
- equate to eigenvalue
97
How to find magnetic quantum number given a wavefunction
Apply Lz operator and equate eigenvalue
98
How to find total angular momentum number given a wavefunction
Apply L^2 operator and compare with eigenvalue
99
Angular Hamiltonian Eigenvalue
L^2 ( theta, Phi)/ (2Mr^2)
100
Hamiltonian for 2 masses connected by rigid rod
2 x Hamiltonian = 2* L^2 ( theta, Phi)/ (2Mr^2) = L^2 ( theta, Phi)/ (Mr^2)
For r, use half of rod length
101
Eigenfunctions of L^2
Spherical harmonics
102
Wavelength mm
Microwaves
103
Correct way to find expectation value from normalised matrix
Sandwich filling
= <Ψl S l Ψ>
(Always check normalised)
1. Apply operator to state
2. Multiply by bra version of state
104
Important to remember when integrating angular momentum
Jacobian
105
How many degrees of freedom in nitrogen molecule?
3x2 = 6
106
How to find expectation value only from superposition of wavefunctions (prefactors and Ψs only) and operator (only defined in terms of Ψ)
1. Check orthonormalisation
2. Apply operator to arbitrary wavefunction Ae^ikx
3. Expand <Ψl S lΨ> in Dirac
4. Cancel, inc as per orthonormality rules
107
Synonymous with “eigenvector”
Eigenfunction, eigenstate, eigenmode
108
Effective potential definition
Sum of repulsive and attractive potentials
109
Effective potential equation
v(eff as a function of r) = V(r) + curly l(curly l + 1) ℏ^2 / (2mr^2)
110
How to sketch Veff graph
1. Find Veff
2. Sub in value for l and sketch
3. Axes ( x in nm on 0.2nm increments up to 1)
4. Centrifugal arises top left - maximim at Veff minimum, quickly asymptotes towards V(r) = 0
5. Coulomb potential arises bottom left - starts at centrifugal slope starts to decline, follows Veff from there getting closer but never touching
111
Calculate commutation in different axes
- end up with 3 terms -> two are +/- and equal magnitude, other cancels as e.g. dx/dy
112
Eigenfunctions of L^2
Spherical harmonics Y (Superscript m, subscript curly l )
113
114
3D harmonic oscillator potential
1/2 m w^2 r^2
115
Rotational energy of single particle
L^2 / 2Mr^2
116
Derive cyclical permutation of Lx,Ly,Lz
Explanation of cross product
117
Angular momentum operator (1D)
Li = - ih (j d/dk - k d/dj)
118
Which angular momentum operators commute?
Only L^2 with individual components
119
What can we say about L^2 and Lz commuting?
Share common eigenfunctions, different eigenvalues
NB neither have radial dependence
120
Ansatz Wavefunction for angular momentum
U (r, θ, Φ) = A (r, θ) e ^ ( (iλΦ) /h)
121
122
123
How to derive magnetic quantum number
1. Ansatz for Lz
U (r, θ, Φ) = A (r, θ) e ^ ( (iλΦ) /h)
2. Apply operator
3. Check boundary conditions
- as angular 0-> 2 π
4. End with e^i2πλ/h = 1
- note true where m = ±0, ±1, ±2….
124
M given X number of curly L states in angular momentum
2X + 1
125
M given X number of curly L states in spin
2X
126
What does ket bra give you?
An operator/ matrix
127
Expected outcome of Stern-Gerlach
Behave like tiny bar magnets due to orbital angular momentum
- point in any direction as force would vary continuously depending on angle of magnetic moment on entering B field
128
Features of Stern Gerlach
- Silver atoms - neutral charge, 1 unpaired electron
- Inhomogenous B field
- Magnetic moment aligned to field
- Split up or down depending on spin
129
Formula for energy shift of a particle in a magnetic field
ΔE = - μ.B = + e/ (2me) L . B
130
Why do we only consider electron’s magnetic moment in Stern Gerlach?
μ is proportional to q/2m (Lhat)
- smallest mass particle has most effect
131
Why does Stern-Gerlach need inhomogeneous fields
Force = - derivative of potential energy
= -∇ (-μ . B)
132
How to find Hamiltonian just for magnetic interaction from magnetic moment
ΔE = - μ.B = + e/ (2me) L . B
- If for example B field points in z-direction, B = Bez (NB second B is just magnitude)
ΔE = - μ.B = + eB/ (2me) L . ez = + eB/ (2me) Lz
133
Write probability of obtaining value, e.g. position A in Dirac notation
P ( X = A) = ll ll^2
134
Correct: If you have eigenvalues and prefactors of an orthonormality basis, what’s the difference between finding probabilities and expectation values?
- Probability -> Prefactor of that state squared
- Expectation value -> Sum of prefactors squared * it’s eigenvalue (Eigenvalues NOT squared and may have different value e.g. spin up and spin down)
135
How to find total energy for electron in magnetic field
1+2
1. Hamiltonian for magnetic interaction
- e.g. B field in z-direction
ΔE = - μ.B = + eB/ (2me) L . ez = + eB/ (2me) Lz
2. Hamiltonian with no B field (Unlm( r, θ, Φ)
136
Spin X eigenvectors
1/√2
[1, 1 ]
[1, -1]
137
Spin Y eigenvectors
1/√2
[1, i]
[1, -i]
138
Spin Z eigenvectors
[1, 0]
[0, 1]
(No prefactor)
139
How to create a given vector lX> as a linear combination of e.g. Sy eigenvectors
1. l X> = [ A ] + [ 0 ] = A up 1/√2 [ 1 ] + A down 1/√2 [ 1 ]
[0 ] + [ B ] [ i ] [ -i ]
Where A and B are l X>’s prefactors, and other column matrices are Sy’s eigenvectors
2. Simultaneous equations reading in rows
3. Sub A down and A up into 1
140
How to expectation value of Sy after being asked to create a given vector lX> as a linear combination of e.g. Sy eigenvectors
Sum of prefactors squared * respective eigenvalues
141
Derive magnetic moment equation for electron
1. I = Q/t
- frequency = 2 π w
- Q = -e for electron
-> I = -e w/(2π)
2. A = A n hat
- A = area vector = πr^2
- n - unit vector normal to area surface
3. μ = I A - sub in 1 and 2
μ = -e w/(2π) * (πr^2)
4. Cancelling ->
μ = -e/2 * w * r^2
142
Derive quantum magnetic moment
1 1. I = Q/t
- frequency = 2 π w
- Q = -e for electron
-> I = -e w/(2π)
2. A = A n hat
- A = area vector = πr^2
- n - unit vector normal to area surface
3. μ = I A - sub in 1 and 2
μ = -e w/(2π) * (πr^2)
4. Cancelling ->
μ = -e/2 * w * r^2
5. Angular momentum L = mwr^2
- sub in and cancel
-> μ = -e/(2me) L hat
143
Value of Hamiltonian due to magnetic moment interaction i.e. energy shift
+ eB/ (2me) * eigenvalue for component of L
E.g. for Lz:
+ eB {ℏm}/ (2me)
144
How does magnetic field E affect electron energy conceptually?
Makes energy of electron (and therefore atom) m-dependent
145
Bohr Magneton in words
Natural unit of magnetic moment for an electron due to its spin or orbital momentum
146
Bohr magneton formula
μb = e ℏ/(2me)
147
Order of matrix multiplication e.g AxB
A row by B column, like bra-ket
- BUT complete every A1 row multiplication in C1 row (B column 1-3), every B1 row in C2 column (B column 1-3) etc
148
Spinor
Spin state as 2D vector
149
Remember when normalising bra-ket
Bra is complex conjugate and a row
150
Value of [Sx, Sy]
ih Sz
151
Define infinite isotopic wel
Particle confined in well with same properties in all directions
E.g. perfectly smooth spherical
152
Difference for harmonic potential energy levels
Equidistant spacing
153
Energy level for FSW, ISW and Harmonic
- FSW = quantised, depends on well depth
- ISW = proportional to n^2
- Harmonic = proportional to n
154
Form of solutions for isotropic square well
Spherical harmonics
155
156
How to create spin operator in arbitrary direction? -> will most likely be give matrix e.g. n = [A,B,C]
1. State n.S_hat_sub_n = n.S_hat
2. Write n formula in terms of S i.e A x_hat becomes A s_x
3. Multiply eigenvector and eigenbasis by each corresponding arbitrary matrix
4. Add all ups and downs to form 1 (2x2) matrix as a combination of Sx, Sy and Sz
157
How to verify a matrix is an eigenvector of an operator matrix Sn
Sn l Up_n > = h/2 l Up_n >
-> h/2 [ Sn matrix ] * matrix being verified l Up_n > = h/2 l Up_n >
1. Expand matrices on left to check equal right
158
How to find probability of measuring observable after cumulative operations e.g. spin up after passing through 1 magnet at angle θ1, THEN second magnet at angle θ2?
- Find eigenstate
- Write arbitrary eigenvector e.g alpha Up vector + Beta down
- Proability is inner product of both spin parts of magnets (product of both alphas)
CAREFUL
- Second is bra, first is ket
159
