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Flashcards in Using Schrodingers Equation Deck (33)
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1
Q

How do you derive the time independent Schrodinger equation?

A

You just add a term for the potential energy - TDSE + V(x) Ψ(x,t). Then substitute in Ψ(x,t) = ϕ(x)p(t), and rearrange

2
Q

What is the time independent Schrodinger equation?

A

-ħ^2/2m * d^2 ϕ/dx^2 + vϕ = Eϕ

3
Q

What is the equation for energy times momentum?

A

iħ * dp/dt = Ep

4
Q

What is the probability density given by for the TISE?

A

|Ψ|^2 = ϕϕ* pp* = |ϕ|^2

5
Q

What is a solution for the TISE?

A

p(t) = exp(-iEt/ħ)

6
Q

What is the infinite potential well problem and how is it solved?

A

A well between x=0 and x=L where the potential inside is 0 and outside in infinity. Need to solve the TISE by setting V(x) = 0, and k^2 = 2mE/ħ^2

7
Q

What is a solution to the TISE for the infinite potential well?

A

ϕ = Asin(kx) + Bcos(kx)

Substitute this into the TISE and see if it works.

8
Q

How do you find the constants for the solution of the TISE?

A

You find the constants using the boundary conditions (x=0 and x=L). To find A, need to use normalisation conditions - integrate A^2 sin^2 (npi*x/L) dx between 0 and L, and set the solution equal to 1. Rearrange to find A.

9
Q

What is the normalised wavefunction for a particle in the infinite potential well?

A

ϕ = (2/L)^(1/2)sin(npix/L)

10
Q

What are the energy levels for a particle in the infinite potential well?

A

En = (ħ^2k^2)/(2m) = (ħ^2n^2*pi^2)/(2mL^2)

11
Q

What is the finite potential well problem?

A

The same as the infinite potential well problem but with finite potential outside the well rather than infinite.

12
Q

How do you solve the finite potential well problem for inside the well?

A

Same as the infinite problem - d^2 ϕ/dx^2 + k^2 ϕ = 0, where k^2 = (2m*E)/(ħ^2)

13
Q

How do you solve the finite potential well problem for outside the well. Can we solve this?

A

Instead of setting V as 0,set it as V0. Then divide through by ħ/2m, and set equal to zero. Then set the constant before ϕ as α.

14
Q

How can we solve the finite potential well problem?

A

Have to match the wave functions inside and outside the well so that they satisfy the boundary conditions: Ψ(x) and dΨ(x)/dx must be continuous at x=0 and x=L.

15
Q

What is the finite potential barrier?

A

Finite potential well flipped upside down with 3 zones - before the barrier, inside the barrier and after the barrier.

16
Q

What are the TISE’s for regions 1 and 3 for the finite potential barrier?

A

1: d^2 ϕ1/dx^2 + (2mE)/(ħ^2) ϕ1 = 0
3: d^2 ϕ3/dx^2 + (2m
E)/(ħ^2) ϕ3 = 0

17
Q

What are solutions for the equations for regions 1 and 3 in the finite potential barrier problem? What do these correspond to?

A

ϕ1 = Aexp(ikx) + Bexp(-ikx)
ϕ3 = Fexp(ikx) + Gexp(-ikx)
k = sqrt(2m*E)/ħ
The A term is the incident wave, the B term is the reflected wave, the F term is the transmitted wave and G is zero since there is no left travelling wave after the barrier.

18
Q

What is the transmission probability for the finite potential barrier?

A

(|ϕ3|^2)/(|ϕ1|^2) = FF/AA

19
Q

What is the TISE for region 2 in the finite potential barrier problem? What is a solution to this?

A

d^2 ϕ2/dx^2 + (2m)/(ħ^2) *(E-V)ϕ2 = 0

Solution is ϕ2 = Cexp(ik’x) + Dexp(-ik’x), with k’ = sqrt(2m(E-V))/ħ

20
Q

What can we say about the wavenumber of region 2?

A

It is imaginary because E is less than V. We can therefore replace -ik’ with k2.

21
Q

How do you find the constants of each region?

A

Use the boundary conditions where ϕ1=ϕ2 and ϕ2 = ϕ3: ik1A - ik1B = -k2C + k2D

and

-k2Cexp(-k2L) + k2Dexp(k2L) = ik3Fexp(ik3L)

22
Q

What is the A/F ratio?

A

A/F = (1/2 + (ik2/4k1))*exp((ik1+k2)L)

23
Q

What is the final equation for the transmission probability?

A

T ∝ exp(-2k2L) ∝ exp((-2Lsqrt(2m(V-E)))/ħ)

24
Q

What does the transmission probability equation mean?

A

That the wave function exponentially decreases within the barrier and then continues at the new amplitude after the barrier.

25
Q

What is scanning tunnelling microscopy?

A

A technique that allows conductive surfaces to be mapped at atomic resolution.

26
Q

What is the transmission probability of tunnelling phenomena?

A

T ~ exp(-(2L*sqrt(2m(V-E)))/ħ)

27
Q

What is an alpha particle?

A

2 protons and 2 neutrons bound together, with a nuclear binding energy of 28.3MeV.

28
Q

What is the Geiger-Nuttall law?

A

ln(λ) = -C1*((Z-2)/sqrt(E)) + C2, where λ is the decay constant, and Eis the energy of emitted alpha particles.

29
Q

What does the potential energy function for an alpha particle look like (interacting with a nucleus of radius R)?

A
  • Potential energy on y axis and distance from centre of nucleus on x axis.
  • Starts off below zero, then shoots up, then decreases by 1/r
30
Q

What is the equation for half life?

A

t(1/2) = ln(2)/λ = τ*ln(2), where τ is the reciprocal of the decay constant λ.

31
Q

What is the equation for number of nuclei remaining at time t (alpha decay)?

A

N(t) = N(0) exp(-λt)

32
Q

What is the approximate range of potential energies on the PE curve?

A

About 30 MeV.

33
Q

What is the equation for potential energy and kinetic energy of alpha particle?

A

Both the same equation - Ek = PE = (2(Z-2)e^2)/(4piϵ0d)