PHY3702
Assignment 3
Due 2025
, Solution to Exercise 6.2
Quantum Mechanics: Concepts and Applications –
Zettili (2nd Ed.)
Exercise 6.2
Problem Statement
A particle of mass m moves in the xy-plane in the potential
1 mω 2 y 2 , 0 < x < a,
V (x, y) = 2
+∞, elsewhere.
(a) Write down the time-independent Schrödinger equation for this particle and reduce
it to a set of familiar one-dimensional equations.
(b) Find the normalized eigenfunctions and the eigenenergies.
Part (a): Time-Independent Schrödinger Equation
Step 1: Write the time-independent Schrödinger equation
The time-independent Schrödinger equation in two dimensions for a particle of mass m
is:
ℏ2 ∂ 2 ψ ∂ 2 ψ
− + 2 + V (x, y)ψ(x, y) = Eψ(x, y).
2m ∂x2 ∂y
Inside the region 0 < x < a, substitute V (x, y) = 21 mω 2 y 2 :
ℏ2 ∂ 2ψ ∂ 2ψ
1
− + 2 + mω 2 y 2 ψ = Eψ.
2m ∂x2 ∂y 2
Outside 0 < x < a, V = +∞, so ψ = 0.
1
, Step 2: Apply separation of variables
Assume the wave function is separable: ψ(x, y) = X(x)Y (y). Substitute into the
Schrödinger equation:
ℏ2 1
− (X ′′ (x)Y (y) + X(x)Y ′′ (y)) + mω 2 y 2 X(x)Y (y) = EX(x)Y (y).
2m 2
Divide by X(x)Y (y):
ℏ2 X ′′ (x) Y ′′ (y)
1
− + + mω 2 y 2 = E.
2m X(x) Y (y) 2
Rearranged:
ℏ2 X ′′ (x) ℏ2 Y ′′ (y) 1
− =E+ − mω 2 y 2 .
2m X(x) 2m Y (y) 2
Since the left-hand side depends only on x and the right-hand side only on y, each side
must equal a constant, say Ex :
ℏ2 X ′′ (x) ℏ2 Y ′′ (y) 1
− = Ex , − + mω 2 y 2 = Ey = E − Ex .
2m X(x) 2m Y (y) 2
Step 3: Identify the one-dimensional equations
- The x-equation:
d2 X 2mEx
+ X = 0, X(0) = X(a) = 0,
dx2 ℏ2
which is the Schrödinger equation for a 1D infinite potential well.
- The y-equation:
ℏ2 d2 Y 1
− 2
+ mω 2 y 2 Y = Ey Y,
2m dy 2
which is the harmonic oscillator Schrödinger equation.
Final Answer for (a) The problem separates into: - A 1D infinite potential well in x -
A 1D harmonic oscillator in y
2
Assignment 3
Due 2025
, Solution to Exercise 6.2
Quantum Mechanics: Concepts and Applications –
Zettili (2nd Ed.)
Exercise 6.2
Problem Statement
A particle of mass m moves in the xy-plane in the potential
1 mω 2 y 2 , 0 < x < a,
V (x, y) = 2
+∞, elsewhere.
(a) Write down the time-independent Schrödinger equation for this particle and reduce
it to a set of familiar one-dimensional equations.
(b) Find the normalized eigenfunctions and the eigenenergies.
Part (a): Time-Independent Schrödinger Equation
Step 1: Write the time-independent Schrödinger equation
The time-independent Schrödinger equation in two dimensions for a particle of mass m
is:
ℏ2 ∂ 2 ψ ∂ 2 ψ
− + 2 + V (x, y)ψ(x, y) = Eψ(x, y).
2m ∂x2 ∂y
Inside the region 0 < x < a, substitute V (x, y) = 21 mω 2 y 2 :
ℏ2 ∂ 2ψ ∂ 2ψ
1
− + 2 + mω 2 y 2 ψ = Eψ.
2m ∂x2 ∂y 2
Outside 0 < x < a, V = +∞, so ψ = 0.
1
, Step 2: Apply separation of variables
Assume the wave function is separable: ψ(x, y) = X(x)Y (y). Substitute into the
Schrödinger equation:
ℏ2 1
− (X ′′ (x)Y (y) + X(x)Y ′′ (y)) + mω 2 y 2 X(x)Y (y) = EX(x)Y (y).
2m 2
Divide by X(x)Y (y):
ℏ2 X ′′ (x) Y ′′ (y)
1
− + + mω 2 y 2 = E.
2m X(x) Y (y) 2
Rearranged:
ℏ2 X ′′ (x) ℏ2 Y ′′ (y) 1
− =E+ − mω 2 y 2 .
2m X(x) 2m Y (y) 2
Since the left-hand side depends only on x and the right-hand side only on y, each side
must equal a constant, say Ex :
ℏ2 X ′′ (x) ℏ2 Y ′′ (y) 1
− = Ex , − + mω 2 y 2 = Ey = E − Ex .
2m X(x) 2m Y (y) 2
Step 3: Identify the one-dimensional equations
- The x-equation:
d2 X 2mEx
+ X = 0, X(0) = X(a) = 0,
dx2 ℏ2
which is the Schrödinger equation for a 1D infinite potential well.
- The y-equation:
ℏ2 d2 Y 1
− 2
+ mω 2 y 2 Y = Ey Y,
2m dy 2
which is the harmonic oscillator Schrödinger equation.
Final Answer for (a) The problem separates into: - A 1D infinite potential well in x -
A 1D harmonic oscillator in y
2
