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Physics Notes 10- 25 - 16 Separation of variables 1D 3D Ψ(x,t) Ψ(x,y,z,t) Prob: (x….x+ Δx) Prob. (x…. x+ Δx, y…..y+Δy, z….z+Δz) | Ψ(x,t)|^2 * Δx (inside a 3-D volume Δτ=ΔxΔyΔz) | Ψ(x,y,z,t)|^2 * Δx* ΔyΔz Require: ∫ |Ψ(x,t)|^2 dx=1 ∫∫∫|Ψ(x,y,z,t)|^2 dxdydz= 3 position operators X,Y,Z (X (^) Ψ )(x,y,z) = x * ϕ(x,y,z) (Y (^) Ψ )(x,y,z) = y * ϕ(x,y,z) (Z (^) Ψ )(x,y,z) = z * ϕ(x,y,z) Momentum P vector form P= (h/i) Px (h/i) d/dx Py (h/i) d/dy Pz (h/i) d/dz H=(Px^2 + Py^2 + Pz^2 )/(2m) + V(x,y,z) = (-h^2 /2m)(d^2 /dx^2 Ψ + d^2 /dy^2 Ψ + d^2 /dz^2 Ψ) + V(x,y,z) Ψ(x,y,z) =H (^) Ψ To Do:
- Find eigenstates H ϕE (x,y,z) = E ϕE(x,y,z)
- Write full solution to Schrödinger equation
V=(x,y,z) 0 if 0 < x < L and 0 < y < L and 0 < z < L and infinity else Separation of Variables: ϕE(x,y,z)= ϕ 1 (x) ϕ 2 (y) ϕ 3 (z) [-h^2 /2m][(d^2 ϕ 1 /dx^2 ) ϕ 2 ϕ 3 + (d^2 ϕ 2 /dy^2 ) ϕ 1 ϕ 3 + (d^2 ϕ 3 /dz^2 ) ϕ 1 ϕ 2 ) + Vϕ 1 ϕ 2 ϕ 3 = E ϕ 1 ϕ 2 ϕ 3 [-h^2 /2m][(1/ϕ 1 (x))(d^2 ϕ 1 /dx^2 ) + (1/ϕ 2 )(d^2 ϕ 2 /dy^2 ) + (1/ϕ 3 )(d^2 ϕ 3 /dz^2 )] + V = E F(x)=E 1 G(y)=E 2 H(z)=E 3 3 Equations (-h^2 /2m)(d^2 Ψ(x)/dx^2 )=Eϕ 1 (x) ϕ 1 (x)= Asin(n πx/L), E 1 =(n^2 π^2 h^2 /2mL^2 ) 0 < x < L, 0 else (-h^2 /2m)(d^2 Ψ(y)/dy^2 )=Eϕ 2 (y) ϕ 2 (x)= Asin(m πy/L),E 2 =(m^2 π^2 h^2 /2mL^2 ) (-h^2 /2m)(d^2 Ψ(z)/dz^2 )=Eϕ 3 (z) ϕ 3 (x)= Asin(k πz/L), E 3 =(k^2 π^2 h^2 /2mL^2 ) Solution: ϕn,m,k(x,y,z) = Asin(n πx/L) sin(m πy/L)sin(k πz/L) Quantum Numbers n,m,k=1,2,3…. En,m,k = (π^2 h^2 /2mL^2 )(n^2 +m^2 +k^2 ) Lowest (ground state) energy E 1 , 1 , 1 = 3 (π^2 h^2 /2mL^2 ) Degeneracy: Same energy for several different states, e.g. E 1 , 1 , 2 = E 1 , 2 , 1 = E 2 , 1 , 1
