Using Schrodingers Equation Flashcards
How do you derive the time independent Schrodinger equation?
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
What is the time independent Schrodinger equation?
-ħ^2/2m * d^2 ϕ/dx^2 + vϕ = Eϕ
What is the equation for energy times momentum?
iħ * dp/dt = Ep
What is the probability density given by for the TISE?
|Ψ|^2 = ϕϕ* pp* = |ϕ|^2
What is a solution for the TISE?
p(t) = exp(-iEt/ħ)
What is the infinite potential well problem and how is it solved?
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
What is a solution to the TISE for the infinite potential well?
ϕ = Asin(kx) + Bcos(kx)
Substitute this into the TISE and see if it works.
How do you find the constants for the solution of the TISE?
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.
What is the normalised wavefunction for a particle in the infinite potential well?
ϕ = (2/L)^(1/2)sin(npix/L)
What are the energy levels for a particle in the infinite potential well?
En = (ħ^2k^2)/(2m) = (ħ^2n^2*pi^2)/(2mL^2)
What is the finite potential well problem?
The same as the infinite potential well problem but with finite potential outside the well rather than infinite.
How do you solve the finite potential well problem for inside the well?
Same as the infinite problem - d^2 ϕ/dx^2 + k^2 ϕ = 0, where k^2 = (2m*E)/(ħ^2)
How do you solve the finite potential well problem for outside the well. Can we solve this?
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.
How can we solve the finite potential well problem?
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.
What is the finite potential barrier?
Finite potential well flipped upside down with 3 zones - before the barrier, inside the barrier and after the barrier.
What are the TISE’s for regions 1 and 3 for the finite potential barrier?
1: d^2 ϕ1/dx^2 + (2mE)/(ħ^2) ϕ1 = 0
3: d^2 ϕ3/dx^2 + (2mE)/(ħ^2) ϕ3 = 0
What are solutions for the equations for regions 1 and 3 in the finite potential barrier problem? What do these correspond to?
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.
What is the transmission probability for the finite potential barrier?
(|ϕ3|^2)/(|ϕ1|^2) = FF/AA
What is the TISE for region 2 in the finite potential barrier problem? What is a solution to this?
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
What can we say about the wavenumber of region 2?
It is imaginary because E is less than V. We can therefore replace -ik’ with k2.
How do you find the constants of each region?
Use the boundary conditions where ϕ1=ϕ2 and ϕ2 = ϕ3: ik1A - ik1B = -k2C + k2D
and
-k2Cexp(-k2L) + k2Dexp(k2L) = ik3Fexp(ik3L)
What is the A/F ratio?
A/F = (1/2 + (ik2/4k1))*exp((ik1+k2)L)
What is the final equation for the transmission probability?
T ∝ exp(-2k2L) ∝ exp((-2Lsqrt(2m(V-E)))/ħ)
What does the transmission probability equation mean?
That the wave function exponentially decreases within the barrier and then continues at the new amplitude after the barrier.
What is scanning tunnelling microscopy?
A technique that allows conductive surfaces to be mapped at atomic resolution.
What is the transmission probability of tunnelling phenomena?
T ~ exp(-(2L*sqrt(2m(V-E)))/ħ)
What is an alpha particle?
2 protons and 2 neutrons bound together, with a nuclear binding energy of 28.3MeV.
What is the Geiger-Nuttall law?
ln(λ) = -C1*((Z-2)/sqrt(E)) + C2, where λ is the decay constant, and Eis the energy of emitted alpha particles.
What does the potential energy function for an alpha particle look like (interacting with a nucleus of radius R)?
- 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
What is the equation for half life?
t(1/2) = ln(2)/λ = τ*ln(2), where τ is the reciprocal of the decay constant λ.
What is the equation for number of nuclei remaining at time t (alpha decay)?
N(t) = N(0) exp(-λt)
What is the approximate range of potential energies on the PE curve?
About 30 MeV.
What is the equation for potential energy and kinetic energy of alpha particle?
Both the same equation - Ek = PE = (2(Z-2)e^2)/(4piϵ0d)
