Derive an expression for the normalized wave function of a particle trapped in an infinite deep potential well.
The Correct Answer and Explanation is :
Normalized Wave Function for a Particle in an Infinite Potential Well
For a particle trapped in a one-dimensional infinite potential well (or box) of width ( L ), the potential ( V(x) ) is defined as:
- ( V(x) = 0 ) for ( 0 \leq x \leq L )
- ( V(x) = \infty ) for ( x < 0 ) and ( x > L )
Derivation of the Wave Function:
The Schrödinger equation for this system is:
[
- \frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} = E \psi
]
Inside the well, the potential energy ( V(x) = 0 ), so the equation simplifies to:
[
\frac{d^2 \psi}{dx^2} = -\frac{2mE}{\hbar^2} \psi
]
Let ( k^2 = \frac{2mE}{\hbar^2} ). Thus, the equation becomes:
[
\frac{d^2 \psi}{dx^2} = -k^2 \psi
]
The general solution to this differential equation is:
[
\psi(x) = A \sin(kx) + B \cos(kx)
]
Boundary conditions dictate that ( \psi(0) = 0 ) and ( \psi(L) = 0 ). Applying these: - ( \psi(0) = 0 ) implies ( B = 0 ) (since ( \cos(0) = 1 ))
- ( \psi(L) = 0 ) leads to ( A \sin(kL) = 0 )
Since ( A \neq 0 ) for a non-trivial solution, ( \sin(kL) = 0 ) which means ( kL = n\pi ) where ( n ) is a positive integer. Hence, ( k = \frac{n\pi}{L} ).
The solutions are therefore:
[
\psi_n(x) = A_n \sin\left(\frac{n\pi x}{L}\right)
]
where ( n = 1, 2, 3, \ldots ).
Normalization:
The wave function must be normalized:
[
\int_0^L |\psi_n(x)|^2 \, dx = 1
]
Substitute ( \psi_n(x) = A_n \sin\left(\frac{n\pi x}{L}\right) ):
[
\int_0^L |A_n|^2 \sin^2\left(\frac{n\pi x}{L}\right) \, dx = 1
]
Using ( \int_0^L \sin^2\left(\frac{n\pi x}{L}\right) \, dx = \frac{L}{2} ), we find ( |A_n|^2 \frac{L}{2} = 1 ), so ( |A_n|^2 = \frac{2}{L} ).
Thus, ( A_n = \sqrt{\frac{2}{L}} ) and the normalized wave function is:
[
\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)
]
This normalized wave function satisfies the boundary conditions, the differential equation, and the normalization condition, fully describing the state of a quantum particle in an infinite potential well.
I’ll now generate an image to visually represent the wave function inside an infinite potential well.
Here’s the visual representation of the quantum wave functions inside a one-dimensional infinite potential well, showing different energy states as sine waves confined within the well. Each wave represents a different quantum state, differentiated by color and wavelength.
