6 Quantum mechanical systems in three dimensions Flashcards
(19 cards)
How do the orbital and magnetic quantum numbers relate?
-l ≤ m ≤ l
What is the Schrodinger equation in 3D?
iℏ∂ψ/∂t = -ℏ2/2M(∂2ψ/∂x2 + ∂2ψ/∂y2 + ∂2ψ/∂z2)
What is the Hamiltonian operator in 3D?
H= -ℏ2/2M ∇2 +V(r,t)
What is the general solution for the wavefunction in 3D?
ψ(r,t) = Φ(r)e-iEt/ℏ
What is the potential for a potential well?
V(r) = 0 inside the well and infinity outside
What is the equation for the potential energy operator in 3D?
V(r) = kxX2/2 + kyY2/2 + kzZ2/2 = Vx(x) + Vy(y) + Vz(z)
Where k is the spring constant
What are the energies for a harmonic oscillator?
Enx = (nx + 1/2) ℏωx
etc for y and z
What is the frequency for a harmonic oscillator?
ω =sqrt(k/M)
What are the spherical coordinates?
x = rsinθcosφ
y = rsinθsinφ
z = rcosθ
What is the equation for the potential of an atom?
V(r) = -Ze2/4πε0r
Why does the power series for a hydrogen-like atom start at 1?
Because for p=0, r=0 and requires a0 = 0
How are the coefficients for the power series found?
ap+1/ap = (p - β)/(p(p+1) - l(l+1))
Fixing a1 or a2
How is the power series normalised?
By terminating the whole sum at a finite term
Set β to n such that p>n = 0
What is the quantum number condition for hydrogen?
l < n
What is the equation for degeneracy?
2l+1
How is the angular momentum quantised?
L=ℏsqrt(l(l+1))
What are the energy levels for hydrogen-like atom?
En = E1/n2
What is the scale factor for energy levels?
Z2
What is the distance of an electron in the Bohr model?
n2a0
