Particle in a box Flashcards
Set up for particle in a box
Similar to free particle but restricted within a box length L
Boundary conditions for particle in a box
Infinite potential energy at the sides, elsewhere is 0
(particle cannot escape)
Equations for boundary conditions potential energy :
V(x<=0) = infinity
V(x=>L) = infinity
V(0<x<L) = 0
Where L is the length of our box
Equation for wavefunction at boundaries and explained
Ψ(x=0) = 0, Ψ(x=L) = 0,
hence we will have a standing wave with n+1 nodes
General wavefunction for standing wave:
Asin(kx+phi)
Steps to solve for E:
Use boundary condition of Ψ at 0 to show phi = 0
Use boundary condition of Ψ at L to show kL = npi, where n = 1,2,3… rearrange for k.
Use standing wave equation and k=k, to get in terms of λ.
Sub in de broigle for λ, to get in terms of p.
Use E =p^2/2m and sub in for p to show quantised
Standing wave equation
k = 2pi/λ
Is energy quantised for particle in a box?
Yes
Steps for calculating A
Use probability density integral with general wavefunction (no phi), between 0 and L.
Take out A and use triganometric identity:
sin^2(x) = (1-cos(2x))/2
Solve for A = (2/L)^1/2
Final wavefunction for particle in a box
Ψ = ((2/L)^1/2)Sin(npix/L)
How to find delta E
Delta E = En+1 - En, and simplify and solve
What happens to delta E as L tends towards infinity, and what does this tell us
Delta E tends towards 0, so for macroscopic L’s energy is unquantised
Zero point energy equal to 0?
No it is when we are in the ground state at n=1
What happens to values calculated when in a 2d box
Energies will be added together with different quantum numbers, and wavefunctions will be multiplied together
What type of molecules can be explained via particle in a box
Long conjugated straight-ish molecules
