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Chapter 2: Problem 38

Consider a particle moving in one dimension under the influence of a potential \(V(x)\). Suppose its wave function can be written as \(\exp [i S(x, t) / \hbar]\). Prove that \(S(x, t)\) satisfies the classical Hamilton-Jacobi equation to the extent that \(\hbar\) can be regarded as small in some sense. Show how one may obtain the correct wave function for a plane wave by starting with the solution of the classical Hamilton-Jacobi equation with \(V(x)\) set equal to zero. Why do we get the exact wave function in this particular case?

Short Answer

Expert verified
The wave function fulfills the Hamilton-Jacobi equation in the classical limit, giving exact solutions for free particles or plane waves.

Step by step solution

01

Express the Wave Function

The wave function is given by \( \psi(x, t) = \exp\left[ \frac{i}{\hbar} S(x, t) \right] \). We need to analyze how this expression relates to the classical Hamilton-Jacobi equation.
02

Consider a Limiting Case for Small \( \hbar \)

Assume \( \hbar \to 0 \) to highlight the classical limit of quantum mechanics where the wave function converges to classical mechanics described by the Hamilton-Jacobi equation.
03

Use the Time-Dependent Schr枚dinger Equation

The Schr枚dinger equation is given by \( i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V(x)\psi \). Substitute \( \psi(x, t) = \exp\left[ \frac{i}{\hbar} S(x, t) \right] \) into this equation.
04

Substitute and Simplify

When you substitute, derive \( \frac{i}{\hbar} \frac{\partial S}{\partial t} \psi = -\frac{\hbar^2}{2m} \left(\frac{i}{\hbar} \frac{\partial S}{\partial x}\right)^2 \psi + V(x)\psi \), then simplify by collecting terms according to powers of \( \hbar \).
05

Recognize the Hamilton-Jacobi Equation

By taking the \( \hbar \to 0 \) limit, obtain the classical equation: \( \frac{\partial S}{\partial t} + \frac{1}{2m} \left( \frac{\partial S}{\partial x} \right)^2 + V(x) = 0 \), which is the Hamilton-Jacobi equation.
06

Solve for a Free Particle Case

For \( V(x) = 0 \), solve the Hamilton-Jacobi equation where \( S(x, t) = Et - px \) and obtain the wave function as \( \psi(x, t) = \exp\left[ \frac{i}{\hbar}(Et - px) \right] = \exp\left[ \frac{i}{\hbar} (kx - \omega t) \right] \).
07

Interpret the Plane Wave Solution

This solution represents a plane wave with \( k = \frac{p}{\hbar} \) and \( \omega = \frac{E}{\hbar} \), corresponding to the classical particles' properties. This aligns with quantum mechanics, providing the exact wave function.

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Key Concepts

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

Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that describes the behavior of matter and energy on very small scales, such as atoms and subatomic particles. This theory revolutionizes our understanding of the microscopic world, offering explanations where classical mechanics falls short.
A fundamental aspect of quantum mechanics is the concept of probabilities, rather than certainties, dictating the state of a system. Particles do not have definite positions or velocities until measured.
Additionally, quantum mechanics introduces the idea of wave-particle duality. Particles exhibit both wave-like and particle-like properties, which is notably captured by concepts such as the wave function. This framework allows us to understand phenomena like superposition and entanglement, which have no classical analogs.
Wave Function
The wave function is a mathematical function that describes the quantum state of a system. It is denoted typically by the symbol \( \psi(x, t) \) and holds all information about the system's state.
In one-dimensional scenarios, the wave function depends on position \(x\) and time \(t\). Physically, the square of the wave function's magnitude, \( |\psi(x, t)|^2 \), represents the probability density of finding a particle at a given position and time.
The nature of the wave function allows it to encapsulate information about probability amplitudes, which are not directly observable but influence the outcomes of measurements.
The wave function's evolution over time is determined by the Schr枚dinger equation, through which it reflects the potential and kinetic energies of the system involved.
Classical Limit
The classical limit refers to the conditions under which quantum mechanical systems begin to resemble classical systems. This transition is often observed when considering large scales or high quantum numbers, where quantum effects become negligible.
The limit is crucial in contexts like the Hamilton-Jacobi equation, where \( \hbar \), the reduced Planck鈥檚 constant, approaches zero. In this scenario, the wave function's behavior translates into classical trajectories of particles.
This link serves as a bridge between the deterministic world of classical physics and the probabilistic nature of quantum mechanics. In essence, when examining macroscopic systems where quantum influences diminish, the system's behavior aligns with classical expectations.
Schr枚dinger Equation
The Schr枚dinger equation is a cornerstone equation in quantum mechanics that governs how wave functions evolve over time. It is an analog to Newton's second law in classical mechanics but for quantum systems.
The time-dependent Schr枚dinger equation is given by:\[ i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V(x)\psi \]Here, \( i \) represents the imaginary unit, \( \hbar \) is the reduced Planck's constant, and \( V(x) \) denotes the potential energy.
This equation allows for the determination of a system's wave function at any point in time, providing insights into properties like energy levels and temporal dynamics.
For complex systems, solving the Schr枚dinger equation can be challenging, but it is essential for making accurate predictions in quantum mechanics. It highlights the dependency of quantum states on both internal properties and external potential landscapes.

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Most popular questions from this chapter

A box containing a particle is divided into a right and a left compartment by a thin partition. If the particle is known to be on the right (left) side with certainty, the state is represented by the position eigenket \(|R\rangle(|L\rangle)\), where we have neglected spatial variations within each half of the box. The most general state vector can then be written as $$ |\alpha\rangle=|R\rangle\langle R \mid \alpha\rangle+|L\rangle\langle L \mid \alpha\rangle $$ where \(\langle R \mid \alpha\rangle\) and \(\langle L \mid \alpha\rangle\) can be regarded as "wave functions." The particle can tunnel through the partition; this tunneling effect is characterized by the Hamiltonian $$ H=\Delta(|L\rangle\langle R|+| R\rangle\langle L|) $$ where \(\Delta\) is a real number with the dimension of energy. a. Find the normalized energy eigenkets. What are the corresponding energy eigenvalues? b. In the Schr枚dinger picture the base kets \(|R\rangle\) and \(|L\rangle\) are fixed, and the state vector moves with time. Suppose the system is represented by \(|\alpha\rangle\) as given above at \(t=0 .\) Find the state vector \(\left|\alpha, t_{0}=0 ; t\right\rangle\) for \(t>0\) by applying the appropriate time-evolution operator to \(|\alpha\rangle\). c. Suppose at \(t=0\) the particle is on the right side with certainty. What is the probability for observing the particle on the left side as a function of time? d. Write down the coupled Schr枚dinger equations for the wave functions \(\langle R| \alpha, t_{0}=\) \(0 ; t\rangle\) and \(\left\langle L \mid \alpha, t_{0}=0 ; t\right\rangle\). Show that the solutions to the coupled Schr枚dinger equations are just what you expect from (b). e. Suppose the printer made an error and wrote \(H\) as $$ H=\Delta|L\rangle\langle R| $$ By explicitly solving the most general time-evolution problem with this Hamiltonian, show that probability conservation is violated.

Consider a particle in one dimension bound to a fixed center by a \(\delta\)-function potential of the form $$ V(x)=-v_{0} \delta(x) $$ where \(v_{0}\) is real and positive. Find the wave function and the binding energy of the ground state. Are there excited bound states?

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.)

Use the WKB method to find the (approximate) energy eigenvalues for the onedimensional simple harmonic oscillator potential \(V(x)=m \omega^{2} x^{2} / 2\).

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 \(t \geq 0\).
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