Answer to Question #126926 in Quantum Mechanics for christopher seebaran

Question

Using the separation of variables technique, derive the time-independent Schrodinger equation (T.I.S.E.) in three dimensions. 

Answer 1

3-D Time-Independent Schrodinger Equation

(−ℏ²/2m)∇²ψ(r)+V(r)ψ(r)=Eψ(r)

Since we are dealing with a 3-dimensional figure, we need to add the 3 different axes into the Schrondinger equation:

−ℏ²/2m(d²ψ(r)/dx²+d²ψ(r)/dy²+d²ψ(r)/dz²)=Eψ(r)

The easiest way in solving this partial differential equation is by having the wavefunction equal to a product of individual function for each independent variable (e.g., the Separation of Variables technique):

ψ(x,y,z)=X(x)Y(y)Z(z)

Now each function has its own variable:

X(x) is a function for variable x only

Y(y) function of variable y only

Z(z) function of variable z only

Now substitute Equation 4 into Equation 3 and divide it by the product: xyz:

d²ψ/dx²=(YZ)d²X/dx²⇒(1/X)d²X/dx²

d²ψ/dy²=(XZ)d²Y/dy²⇒(1/Y)d²Y/dy²

d²ψ/dz²=(XY)d²Z/dz²⇒(1/Z)d²Z/dz²

E is an energy constant, and is the sum of x, y, and z. For this to work, each term must equal its own constant. For example,

(d²X/dx²⁽+(2m/ℏ²)ε_xX=0

Now separate each term in Equation to equal zero:

(d²X/dx²)+(2m/ℏ²)ε_xX=0

(d²Y/dy²)+(2m/ℏ²)ε_yY=0

(d²Z/dz²)+(2m/ℏ²)ε_zZ=0

Now we can add all the energies together to get the total energy:

E_x+E_y+E_z =E

Now the equations are very similar to a 1-D box and the boundary conditions are identical, i.e.,

n=1,2,..∞

Use the normalization wavefunction equation for each variable:

ψ(x)={(√2/L_x)sinnπ_xL_x } if 0≤x≤L0if L<x<0

Normalization wavefunction equation for each variable

X(x)=(√2/L_x)sin(n_xπ_x/L_x)

Y(y)=(√2/L_y)sin(n_yπ_y/L_y)

Z(z)=(√2/L_z)sin(n_zπ_z/L_z)

The limits of the three quantum numbers

n_x=1,2,3,...∞

n_y=1,2,3,...∞

n_z=1,2,3,...∞

For each constant use the de Broglie Energy equation:

ε_x=n_x²h²/8mL_x²

with n_x=1,2,3...∞

Do the same for variables n_y and n_z. Combine wave functions for all axis , So overall wavefunctions inside a 3D box is

ψ(r)=(√8/L)sin(n_xπ_x/L_x)sin(n_yπ_y/L_y)sin(n_zπ_z/L_z)