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Chapter 2: Problem 28

A particle in one dimension is trapped between two rigid walls: $$ V(x)=\left\\{\begin{array}{lll} 0 & \text { for } & 0L . \end{array}\right. $$ At \(t=0\) it is known to be exactly at \(x=L / 2\) with certainty. What are the relative probabilities for the particle to be found in various energy eigenstates? Write down the wave function for \(t \geq 0\). (You need not worry about absolute normalization, convergence, and other mathematical subtleties.)

Short Answer

Expert verified
Particle is only in odd energy eigenstates; wave function is superposition of these states for \(t \geq 0\).

Step by step solution

01

Understanding the System

The potential function describes a particle in a one-dimensional box of length \(L\) with impenetrable walls at \(x = 0\) and \(x = L\). Inside this box \((0 < x < L)\), the potential \(V(x) = 0\) and outside this region, \(V(x) = \infty\). This means the particle is confined to the region \(0 < x < L\).
02

Identifying the Energy Eigenstates

For a particle in a box, the energy eigenstates are described by the wave functions \( \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right) \) for \( n = 1, 2, 3, \ldots \). These functions are solutions to the Schr枚dinger equation in the box region \([0, L]\).
03

Wave Function at Initial Time

At \(t = 0\), the particle is found at \(x = L/2\) with certainty, which can be written as a delta function \( \delta(x - L/2) \). The delta function corresponds to a strong localization at a single point.
04

Expanding Initial Wave Function in Terms of Energy Eigenstates

The initial wave function \( \delta(x - L/2) \) needs to be expanded as a sum of energy eigenstates: \[ \delta(x - L/2) = \sum_{n=1}^\infty c_n \psi_n(x) \]To find \(c_n\), we use the orthogonality of the eigenstates:\[ c_n = \int_0^L \delta(x-L/2) \psi_n(x) \, dx = \psi_n(L/2) \].
05

Calculating Coefficients for Expansion

The coefficient \( c_n = \psi_n(L/2) = \sqrt{\frac{2}{L}} \sin\left(\frac{n \pi L/2}{L}\right) = \sqrt{\frac{2}{L}} \sin\left(\frac{n \pi}{2}\right) \). The values of \( c_n \) depend on the parity of \( n \):- For odd \( n \), \( \sin\left(\frac{n\pi}{2}\right) = \pm 1 \), hence positive or negative \( \sqrt{\frac{2}{L}} \).- For even \( n \), \( \sin\left(\frac{n\pi}{2}\right) = 0 \), hence \( c_n = 0 \).
06

Determining Relative Probabilities

The relative probability of finding the particle in an eigenstate \( n \) is determined by \( |c_n|^2 \). Thus, for odd \( n \), \(|c_n|^2 = \frac{2}{L}\) and for even \( n \), \(|c_n|^2 = 0\). The particle is therefore only found with a non-zero probability in odd energy eigenstates.
07

Wave Function for \( t \geq 0 \)

The wave function for \( t \geq 0 \) is a superposition of time-evolving energy eigenstates:\[ \Psi(x, t) = \sum_{\substack{n=1 \text{n odd}}}^\infty c_n \psi_n(x) e^{-iE_nt/\hbar} \]where energy eigenvalues are \( E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} \). Only odd terms are present in the sum because the even coefficients are zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Particle in a Box
The concept of a "particle in a box" is fundamental in quantum mechanics and describes a particle that is free to move within a certain region, but is confined by infinite potential barriers at the edges. This scenario is typically considered in one dimension to simplify the mathematics involved. The potential energy function, as described in the original exercise, is zero inside the box ( 0 < x < L ) and infinite outside the box. This implies that the particle cannot exist outside these boundaries.

Key characteristics of a particle in a box include:
  • The particle is completely free within the limits but cannot escape the box due to the infinite potential at the boundaries.

  • Only certain wave functions (solutions to the Schr枚dinger equation) are allowed inside the box.

  • The notion of quantization is introduced, where only discrete energy levels are possible for the particle.

This simple model helps illustrate the wave nature of particles and lays the groundwork for understanding more complex systems in quantum mechanics.
Energy Eigenstates
In the context of quantum mechanics, energy eigenstates are specific states of a system in which the energy is well-defined. For a particle in a box, these states are described by distinct wave functions that solve the time-independent Schr枚dinger equation. Inside the box, the energy eigenstates are represented by sinusoidal wave functions that satisfy boundary conditions.Each eigenstate corresponds to a quantized energy level, denoted by an integer quantum number \( n \). The formula for these energy levels is expressed as \( E_n = \frac{n^2 \, \pi^2 \, \hbar^2}{2mL^2} \), where \( m \) is the mass of the particle, \( L \) is the length of the box, and \( \hbar \) is the reduced Planck constant.

Some key points about energy eigenstates are:
  • Each eigenstate corresponds to a distinct wave function \( \psi_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{n\pi x}{L}) \).

  • Only certain states, or wave functions, are allowed based on boundary conditions.

  • Energy levels become higher as \( n \) increases, with larger gaps between higher energy states.

This quantization of energy is a core concept in quantum mechanics and shows how particles behave differently at the quantum level compared to classical physics.
Wave Functions
Wave functions are mathematical descriptions of the quantum state of a system. They express the probability amplitude for a particle's position and momentum. For a particle in a box, wave functions determine where the particle is likely to be found within the box.

The wave functions for energy eigenstates in a box are given by:
  • \( \psi_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{n\pi x}{L}) \)
These functions have specific characteristics:
  • They are sinusoidal, indicating a wave-like behavior of the particle within the box.

  • They must be zero at the walls of the box, respecting the boundary conditions.

  • Nodes, or points where the wave function is zero, occur within the box and are related to the energy level \( n \).

The probability density \( |\psi(x)|^2 \) tells us the likelihood of finding the particle at a given position \( x \). It's important to note that this probability is not uniformly distributed but rather has maxima and minima, reflecting the wave nature of quantum particles.
Schr枚dinger Equation
The Schr枚dinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. There are two forms: the time-independent Schr枚dinger equation, which is used to determine energy eigenstates; and the time-dependent Schr枚dinger equation, which describes the evolution of wave functions over time.

In the case of a particle in a box, the time-independent Schr枚dinger equation is solved to find the allowed energy levels and associated wave functions.
  • The general form is: \( -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + V(x)\psi = E\psi \).
For regions of constant potential, the solutions are sinusoidal functions inside the box. The time-dependent Schr枚dinger equation introduces a time evolution factor \( e^{-iE_nt/\hbar} \) to these solutions.Key concepts involving the Schr枚dinger equation include:
  • It governs the dynamics of quantum systems, allowing prediction of future behavior.

  • Solutions must comply with boundary conditions, leading to quantization of energy and discrete eigenstates.

The Schr枚dinger equation establishes the framework for understanding quantum phenomena, as it bridges the behavior of particles with wave mechanics.

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Most popular questions from this chapter

Make the definitions $$ J_{\pm} \equiv \hbar a_{\pm}^{\dagger} a_{\mp}, \quad J_{z} \equiv \frac{\hbar}{2}\left(a_{+}^{\dagger} a_{+}-a_{-}^{\dagger} a_{-}\right), \quad N \equiv a_{+}^{\dagger} a_{+}+a_{-}^{\dagger} a_{-} $$ where \(a_{\pm}\)and \(a_{\pm}^{\dagger}\) are the annihilation and creation operators of two independent simple harmonic oscillators satisfying the usual simple harmonic oscillator commutation relations. Also make the definition $$ \mathbf{J}^{2} \equiv J_{z}^{2}+\frac{1}{2}\left(J_{+} J_{-}+J_{-} J_{+}\right) $$ Prove $$ \left[J_{z}, J_{\pm}\right]=\pm \hbar J_{\pm}, \quad\left[\mathbf{J}^{2}, J_{z}\right]=0, \quad \mathbf{J}^{2}=\left(\frac{\hbar^{2}}{2}\right) N\left[\left(\frac{N}{2}\right)+1\right] $$

Consider the spin-precession problem discussed in the text. It can also be solved in the Heisenberg picture. Using the Hamiltonian $$ H=-\left(\frac{e B}{m c}\right) S_{z}=\omega S_{z} $$ write the Heisenberg equations of motion for the time-dependent operators \(S_{x}(t)\), \(S_{y}(t)\), and \(S_{z}(t)\). Solve them to obtain \(S_{x, y, z}\) as functions of time.

Consider a particle subject to a one-dimensional simple harmonic oscillator potential. Suppose at \(t=0\) the state vector is given by $$ \exp \left(\frac{-i p a}{\hbar}\right)|0\rangle, $$ where \(p\) is the momentum operator, \(a\) is some number with dimension of length, and the state \(|0\rangle\) is the one for which \(\langle x\rangle=0=\langle p\rangle\). Using the Heisenberg picture, evaluate the expectation value \(\langle x\rangle\) for \(t \geq 0\).

Consider a particle in three dimensions whose Hamiltonian is given by $$ H=\frac{\mathbf{p}^{2}}{2 m}+V(\mathbf{x}) $$ By calculating \([\mathbf{x} \cdot \mathbf{p}, H]\) obtain $$ \frac{d}{d t}\langle\mathbf{x} \cdot \mathbf{p}\rangle=\left\langle\frac{\mathbf{p}^{2}}{m}\right\rangle-\langle\mathbf{x} \cdot \nabla V\rangle . $$ To identify the preceding relation with the quantum-mechanical analogue of the virial theorem it is essential that the left-hand side vanish. Under what condition would this happen?

A particle with mass \(m\) moves in one dimension and is acted on by a constant force \(F\). Find the operators \(x(t)\) and \(p(t)\) in the Heisenberg picture, and find their expectation values for an arbitrary state \(|\alpha\rangle\). Use \(\langle x(0)\rangle=x_{0}\) and \(\langle p(0)\rangle=p_{0}\). The result should be obvious. Comment on how to do this problem in the Schr枚dinger picture, but do not try to work it through.
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