6. QM in 3D Flashcards
Why is the 1D harmonic potential limited (be specific)?
Because simple diatomic molecules are 3D
- They cacn rotate and stretch
What coordinate system do we use for diatomic molecules in 3D?
Spherical coordinates
State the spherical volume element
dV = r^2 sin(θ)dθdϕdr
State the normalisation condition
The integral everywhere of the modulus squared of the wave function is equal to 1
See page 3 of the document for the normalisation calculation for the wave function
Do it lol
What can we say about angular momentum?
It is quantised - number of values and states is restricted
What is the equation for angular momentum?
L = r x p
State the z component for the angular momentum
L_z = xp_y - yp_x
Translate from Cartesian coordinates to spherical
r = (x,y,z) = (r,θ,ϕ)
Describe the components of the spherical coordinates and their ranges
r - distance from the origin (0 to infinity)
θ - Polar angle (0 to pi)
ϕ - Azinuthal angle (0 to 2 pi)
What type of potential are we only interested in?
Central potential
Mathematically, how are central pontentials described?
V(r) not V(r) or V(r,θ,ϕ)
What is m_l ?
The magnetic quantum number
What is l?
The orbital quantum number
State how we separate the wavefunction u(r)
u(r) = u(r,θ,Φ) = R(r) (Θ(θ) Φ(ϕ)
What can we say about the total energy of the wave funciton for a central potential?
It only depends on the radial equation because it is the only one involting V(r) following the separation of variables
What can we say about the solution of the potential?
V(r) is proportional to 1/r
Describe the proposed solution for solving Φ(ϕ)
Try Φ = A exp(im_lϕ)
Φ’’ + (m_l)^2 Φ = 0
What conditions can we put on Φ which is part of the wavefunction?
- Φ is continuous
- u(r) is allowed to be complex
- Φ is single valued -> doesn’t change around 2pi (it is well behaved)
