A Particle In The Infinite Square Well Has The Initial Wave Function (x,0) = {Ax, 0 < X < A/2{A(a-x),

Introduction

A particle in the infinite square well has the initial wave function (x,0) = {Ax, 0 < x < A/2; A(a - x), A/2 < x < A}. This initial wave function describes a quantum particle confined within a one-dimensional potential well, where the potential is zero inside the well (from x=0 to x=A) and infinite outside this region. The shape of the initial wave function is piecewise linear, increasing from zero at x=0 to a maximum at x=A/2, then decreasing back to zero at x=A. Understanding the evolution of this wave function over time requires expanding it in terms of the well's eigenstates, calculating the coefficients for this expansion, and analyzing the time-dependent behavior.

This article explores the mathematical formulation, the process of decomposing the initial state into eigenstates, and the physical interpretation of the wave function's evolution within the well. We will also discuss the implications for probability densities and expectation values, providing a comprehensive understanding of the system.

The Infinite Square Well and Its Eigenstates

Potential and Boundary Conditions

In the infinite square well, the potential V(x) is defined as:

• V(x) = 0 for 0 < x < A


• V(x) = ∞ for x ≤ 0 or x ≥ A

These boundary conditions enforce that the wave function must vanish at the walls:

• ψ(0, t) = 0


• ψ(A, t) = 0

This leads to discrete energy eigenstates characterized by quantum numbers n = 1, 2, 3, ...

Eigenfunctions and Eigenvalues

The normalized stationary eigenfunctions for the infinite square well are well-known:

\[
\phi_n(x) = \sqrt{\frac{2}{A}} \sin \left( \frac{n \pi x}{A} \right), \quad n=1,2,3,\dots
\]

with corresponding energy eigenvalues:

\[
E_n = \frac{n^2 \pi^2 \hbar^2}{2 m A^2}
\]

where:

• \( \hbar \) is the reduced Planck's constant,


• \( m \) is the mass of the particle,


• \( A \) is the width of the well.

Any initial wave function \( \psi(x, 0) \) can be expanded as:

\[
\psi(x, 0) = \sum{n=1}^\infty cn \phi_n(x)
\]

where the coefficients \( c_n \) are determined by projecting the initial wave function onto each eigenstate.

Initial Wave Function Analysis

Definition of the Wave Function

The initial wave function is given as a piecewise linear function:

\[
\psi(x,0) =
\begin{cases}
A x, & 0 < x < \frac{A}{2} \\
A \left( a - x \right), & \frac{A}{2} < x < A \\
0, & \text{elsewhere}
\end{cases}
\]

where \( A \) is a normalization constant and \( a \) is a parameter that influences the shape. To proceed, we need to determine \( A \) and \( a \) such that \( \psi(x,0) \) is normalized.

Normalization of the Wave Function

The normalization condition requires:

\[
\int_0^A |\psi(x,0)|^2 dx = 1
\]

Calculating this integral:

\[
\int0^{A/2} (A x)^2 dx + \int{A/2}^A [A (a - x)]^2 dx = 1
\]

which simplifies to:

\[
A^2 \left( \int0^{A/2} x^2 dx + \int{A/2}^A (a - x)^2 dx \right) = 1
\]

Computing the integrals:

• First integral:

\[
\int_0^{A/2} x^2 dx = \frac{1}{3} \left( \frac{A}{2} \right)^3 = \frac{A^3}{24}
\]

• Second integral:

Let’s evaluate:

\[
\int_{A/2}^A (a - x)^2 dx
\]

Substitute \( u = a - x \), then \( du = - dx \), and when \( x = A/2 \), \( u = a - A/2 \); when \( x = A \), \( u = a - A \).

The integral becomes:

\[
\int{u = a - A}^{a - A/2} u^2 (-du) = \int{a - A/2}^{a - A} u^2 du
\]

which yields:

\[
\frac{1}{3} u^3 \Big|_{a - A}^{a - A/2} = \frac{1}{3} \left[ (a - A/2)^3 - (a - A)^3 \right]
\]

Putting it all together:

\[
A^2 \left( \frac{A^3}{24} + \frac{1}{3} \left[ (a - A/2)^3 - (a - A)^3 \right] \right) = 1
\]

From here, \( A \) and \( a \) can be chosen or determined based on normalization constraints.

Calculating Expansion Coefficients \( c_n \)

Projection onto Eigenstates

The expansion coefficients are given by:

\[
cn = \int0^A \phi_n^(x) \psi(x,0) dx
\]

Since the eigenfunctions are real, this simplifies to:

\[
cn = \sqrt{\frac{2}{A}} \left[ \int0^{A/2} x \sin \left( \frac{n \pi x}{A} \right) dx + \int_{A/2}^A (a - x) \sin \left( \frac{n \pi x}{A} \right) dx \right]
\]

Calculating these integrals involves integration by parts or lookup of standard integrals. The process is as follows:

• For each integral, express the integrand as the product of a polynomial and sine function.


• Use integration by parts to evaluate:


• \( \int x \sin(kx) dx \),


• \( \int (a - x) \sin(kx) dx \).

The resulting \( c_n \) coefficients encapsulate how the initial wave function decomposes into the eigenstates.

Analytical or Numerical Approach

Given the piecewise linear form, explicit formulas for \( c_n \) may be complex. Therefore, practical calculations often involve numerical integration:

• Discretize the domain,


• Evaluate the integrand at sufficient points,


• Use numerical methods such as Simpson’s rule or Gaussian quadrature.

This approach provides approximate \( c_n \) values, which can then be used to analyze the time evolution.

Time Evolution of the Wave Function

General Expression

The wave function at any time \( t \) can be reconstructed as:

\[
\psi(x, t) = \sum{n=1}^\infty cn \phin(x) e^{-i En t / \hbar}
\]

where:

• \( c_n \) are the expansion coefficients,


• \( \phi_n(x) \) are the eigenfunctions,


• \( E_n \) are the eigenenergies.

This superposition encapsulates the quantum dynamics within the well.

Physical Interpretation

The wave function's time evolution leads to phenomena such as:

• Quantum interference among eigenstates,


• Wave packet spreading,


• Revival and fractional revival patterns.

The initial linear shape influences the distribution of energy eigenstates, affecting how quickly the wave function disperses and how the probability density evolves.

Probability Density and Expectation Values

Probability Density

The probability density at time \( t \) is:

\[
|\psi(x, t)|^2 = \left| \sum{n=1}^\infty cn \phin(x) e^{-i En t / \hbar} \right|^2
\]

This quantity reveals where the particle is most likely to be found at any given moment.

Expectation Values

Key expectation values include:

• Position:

\[
\langle x \rangle (t) = \int_0^A x |\psi(x, t)|^2 dx
\]

• Momentum:

\[
\langle p \rangle (t) = \int_0^A \psi^(x, t) \left( -i \hbar \frac{\partial}{\partial x} \right) \psi(x, t) dx
\]

These provide insights into the particle's average position and momentum over time.

Implications and Applications

Understanding the evolution of this particular