Lecture 2 Flashcards
(78 cards)
1
Q
Why can we represent particles with wave equation?
A
They have “wave-like” nature
2
Q
What assumption do we make about the eternal potential?
A
That it is conservative
- V (r) ≠ V (r,t)
3
Q
What is the equation of motion for a matter-wave?
A
, i ℏ δ Ψ (r,t) / δt = - ℏ^2 / (2m) ∇ ^2 Ψ (r,t) + V (r) Ψ (r,t)
4
Q
What can the Wavefunction equation be used for?
A
Finding energy levels of any storm or molecule
5
Q
How do you separate a wavefunction into space and time dependent parts?
A
Ψ (r,t) = Ψ (r) Φ (t)
OR
Ψ (r,t) = A e ^ i(k.r - wt) = A e^ ik.r e^ -iwt
6
Q
What is 1/ i
A
1/i = - i^2 / i = - i
7
Q
What is d/d Φ = K Φ(t) where K = arbitrary constant?
A
- Derivative of Φ = constant * Φ ——> Φ = exp^ Kt
(Assuming derivative wrt to t, otherwise would be other variable)
8
Q
How to derive TIWE?
A
- Divide wavefunction into time and space dependent parts
- Use separation of variables to make each part = E
- Solve time dependent part
- Notice exponent resembles time dependent part of normal wave (e ^ -iwt) and deduce that E = energy
- Solve space part
9
Q
What is the TIWE?
A
Not a wave equation as no time dependence, instead an energy eigenvalue equation
10
Q
What is the Hamiltonian operator?
A
- ℏ ^2 / (2m) ∇ ^2 + V(r)
11
Q
What is the momentum operator?
A
- i ℏ ∇
12
Q
What is V(x) for a free particle?
A
Ψ (r,t) = A e ^ I’ll
13
Q
What is a solution to TIWE for free particle?
A
Ψ (r,t) = A e ^ ikx
14
Q
How to find energy for free particle?
A
- Use TIWE
- V = 0 as free particle
- Sub in solution Ψ (r,t) = A e ^ ikx
- Complete any differentials to obtain expression for E
15
Q
What is the energy for a free particle?
A
E = ℏ^2 k^2 / (2m)
16
Q
What is the value for potential for ISW?
A
V (x) = 0 when [X] </= a
V(x) = infinity when [x] >a
17
Q
What is the width of the ISW?
A
2a
18
Q
How many parts do you split the ISW problem into?
A
3
- Outisde box, [x] >a
- At boundary [x] = a
- Inside well [x] <a
19
Q
What is the energy outside the box in an ISW?
A
Ψ ( [x] > a ) = 0
- Assume E is never infinite
- Can only ensure this if Ψ (x) = 0
20
Q
What is the energy at the boundary of the box in an ISW?
A
- Assume Ψ ( [x] = a ) is infinite at the boundary
- When check energy, assume Ψ ( [x] = a ) is infinite and second derivative is undefined
- Cannot have undefined energy so Ψ ( [x] = a ) = 0
21
Q
What is the energy inside the box in an ISW?
A
- Most general solution = Ψ (x) = Ce^ikx + De ^ -ikx or A sin (kx) + B cos (kx) (trig form easier here)
- Need Ψ (x) = 0 —-> trivial solution is A = B = 0, meaning Ψ (x) = 0 and no particle in box
- Either A = 0 and cos (ka) = 0 (ka = n* π / 2 for n odd) or B = 0 and sin (ka) =0 (ka = n π/2 for n even)
- Ψ odd(x) = B cos (n π x/ 2a); Ψeven (x) = A sin (n π x/ 2a)
- Sub original wave equation ansatz into TIWE and get H^2k^2/2m Ψ = E Ψ (works for odd and even)
- As ka = n π x/ 2 -» k = n π x/ 2a -»> sub into above
22
Q
What is En?
A
Energy eigenvalues for the eigenfunctions
23
Q
What are the nature of the predicted energy levels for a free particles?
A
Discrete
24
Q
What does confinement of a free particle lead to?
A
Quantised energy
25
What are the energy eigenvalues proportional to?
1/a^2. ->>> tighter confinement leads to higher energies
26
How to find the spacing between energy levels?
Use energy solution for inside ISW and check for difference between n and n+1
- En = ℏ^2/ 2m * (π / 2a)^2 * (2n + 1)
27
Why are gaps between energy levels normally not visible?
ℏ^2 tiny
28
What happens to energy levels with increasing mass?
Spacing between levels decreased and starts to look continuous
29
What is the correspondence principle?
Massive objects do not look quantum
30
What happens as a gets large?
Energy starts to look continuous i.e. smoothly transitions to being like free particle
31
What is the lowest energy?
E1 = ℏ^2 π ^2/ 8ma^2
- E 0 ->>> Ψ (x) = 0 i.e. no particle
32
What is the ground state?
The lowest energy a particle in a box can have aka zero point energy
33
How can a particle have zero kinetic energy but not be at rest?
Ground state energy is non-zero - essentially zero in frame, still some motion though KE = 0
34
What does the Energy eigenfunction look like?
- When n = odd, symmetric about x=0
- When n = even, anti-symmetric about x0=0
35
What is positive parity?
Wavefunction symmetric about x= 0
- unchanged when spatial coordinates flipped
36
What is negative parity?
Wavefunction anti-symmetric about x= 0
- signs change when spatial coordinates flipped
37
When does energy associated with wavefunctions increase?
With increasing numbers of nodes
38
How many antinodes in the nth wavefunction?
N
39
Which values of E do we consider for the FSW and why?
E < Vo so particle sits in well
40
What is the energy outside the well for the FSW?*** needs completion
- Rearrange with Hamiltonian so second derivative of Ψ + (Vo - E ) Ψ = 0
- General solution = Ψ (x) - Ce^ax + De^-ax
- Check limits
- When x>0, as x -> ∞, e ^ α x ∞, so Ψ (x) ->>> ∞ ( not physical, C = 0)
- When x < 0, as x -> -∞, e^ - αx ->> ∞, so Ψ (x) -> ∞ ( not physical, D =0)
- Sub in Ce^ax into derived Hamilton
- Ce^ax cancel and solve for α leaving as +ve square root
41
What is the energy at the boundary of the well for the FSW?***** not finished
- Is now clear Ψ (x=a) is finite to solve wavefunction equation ( same as x
42
What is the energy inside the well for the FSW?*** not finished
- Is now clear Ψ (x
43
What limits the number of solutions to the FSW?
The fact that [k/ko]
44
What is the minimum number of FSW solutions?
1 as long as E
45
When are there more possible solutions for k equations of the FSW?
- Large ko (i.e large Vo)
- Small a
46
What are the black circles on FSW graphs?
Solutions
47
What happens to FSW when you set ko to be a larger number?
Potential well gets deeper
48
What do wavefunctions of FSW look like?
- Wavefunction extends outside box
49
What are useful physical properties of quantum tunnelling?
Radioactive alpha-decay and scanning electron microscopy
50
How do wavefunctions behave outside the FSW box?
- Extends with exponential decay as e^ -α x
51
What is the relative magnitude of α ?
Very large as ℏ^2 is very small, unless mass is very low
52
How to derive tunnelling problem?
- Finite potential barrier Vo and finite width b, particle travels L to R
- 1: Free space where V(x) = 0, Incident particle
- Ψ (x) = Ae^ikx + Be^-ikx
-2: Within well
- Ψ (x) = Ce^ α x + De^-α x
- α = SQRT ( 2m* (Vo-E) / ℏ^2
- cannot consider limits
- 3: Transmitted particle, Free space where V(x) = 0
- Neglect solutions travelling R to L
- Ψ (x) = Fe^ikx
- Create equations using x=b and x=0 and solve simultaneously for F
- [F]^2/[A]^2 = ( 16 E (Vo-E) ) / Vo^2 all * e ^(-2 αb)
53
What does [F]^2/[A]^2 represent?
The tunnelling probability, or fraction of particles getting through barrier
54
What can cause particle tunnelling to fall?
Higher barrier height and width
55
What is the dominant term in tunnelling probability?
- e^-2αb
Ab is a function of m, Vo-E, / h^2 and b
56
How does k change in quantum tunnelling?
Constant because energy isn’t lost
57
What is the potential for harmonic potential?
V ∝ x^2
58
What happens in harmonic potential when x is small?
Region around any potential minima is V(x) 1/2 s x^2
59
How to estimate V(x) in vicinity of x =0 in harmonic potential?
- Taylor expanding at x=0
- V(0) = 0
- 1st derivative = 0 (minimum)
- 2nd derivative = x^2/2
- 3rd derivative = 0
60
What is the force acting on a particle in harmonic potential?
As SHO, use Hooke’s Law
F (X) = -dV/dx = -sx
61
Potential energy for harmonic oscillator
1/2 mw^2 x^2
62
Equation of motion for harmonic oscillator
D^2x/ dt^2 = -w^2x
63
How to obtain Hamilton for harmonic potential?
H = KE + PE
H = - h^2 /2m + 1/2 mw^2 x^2
64
How to rescale position variable in Harmonic Oscillator to solve wavefunction equation?
Y^2 = mwx^2 / h
- Squareroot
65
How to rescale energy in Harmonic Oscillator to solve wavefunction equation?
E = 2E/ hw
66
What is the solution of the quantum harmonic oscillator wavefunction?
D^2 Ψ(y) /d2y + (E-y^2) Ψ (y) = 0
- this is rescaled energy
67
What is the asymptotic solution?
- Solution of rescaled energy (QHO)
- Ensure x ± ∞ as y ± ∞
- As y ∝ x and w ∝ 1/E, E must remain small as x -> ∞
- I.e. as y -> ∞, y >> E
- D^2 Ψ ∞(y) /d2y = y^2 Ψ ∞(y)
68
What does the solution to the asymptotic solution look like?*****
69
What is the non-asymptotic solution?*****
70
Define recursion relation
A series or sequence where each term is a function of the previous
71
What is the energy of the QHO?
En = ( n + 1/2) ℏ w
72
What is the lowest energy of the QHO?
E0 = 1/2 ℏw
73
What is the spacing between QHO energy levels?
ℏw
74
Difference between TDWE and TIWE?
Both have ℏ^2/2m Ψ and V (r) Ψ
- TDWE = i ℏ d Ψ (r,t) and all Ψ have r and t dependence
- TIVE = E Ψ (r) and all Ψ have r dependence only
75
76
What does F squared over A squared depict?
Tunnelling probability
77
What is dominant term in tunnelling probability
E^-2ab
78
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