A particle of mass m, in an infinite potential well of...

A particle of mass $m$, in an infinite potential well of length $a$, has the following initial wave function at $t=0$ : $$\psi(x, 0)=\sqrt{\frac{3}{5 a}} \sin \left(\frac{3 \pi x}{a}\right)+\frac{1}{\sqrt{5 a}} \sin \left(\frac{5 \pi x}{a}\right),$$ and an energy spectrum $E_{n}=-\hbar^{2} \pi^{2} n^{2} /\left(2 m a^{2}\right)$. Find $\psi(x, t)$ at any later time $t$, then calculate $\frac{\partial \rho}{\partial t}$ and the probability current density vector $\vec{J}(x, t)$ and verify that $\frac{\partial \rho}{\partial t}+\vec{\nabla} \cdot \vec{J}(x, t)=0 .$ Recall that $\rho=\psi^{*}(x, t) \psi(x, t)$ and $\vec{J}(x, t)=$ $\frac{i \hbar}{2 m}\left(\psi(x, t) \vec{\nabla} \psi^{*}(x, t)-\psi^{*}(x, t) \vec{\nabla} \psi(x, t)\right) .$

A particle of mass $m$, in an infinite potential well of length $a$, has the following initial wave function at $t=0$ :
$$\psi(x, 0)=\sqrt{\frac{3}{5 a}} \sin \left(\frac{3 \pi x}{a}\right)+\frac{1}{\sqrt{5 a}} \sin \left(\frac{5 \pi x}{a}\right),$$
and an energy spectrum $E_{n}=-\hbar^{2} \pi^{2} n^{2} /\left(2 m a^{2}\right)$. Find $\psi(x, t)$ at any later time $t$, then calculate $\frac{\partial \rho}{\partial t}$ and the probability current density vector $\vec{J}(x, t)$ and verify that $\frac{\partial \rho}{\partial t}+\vec{\nabla} \cdot \vec{J}(x, t)=0 .$ Recall that $\rho=\psi^{*}(x, t) \psi(x, t)$ and $\vec{J}(x, t)=$
$\frac{i \hbar}{2 m}\left(\psi(x, t) \vec{\nabla} \psi^{*}(x, t)-\psi^{*}(x, t) \vec{\nabla} \psi(x, t)\right) .$

Key Concepts

Probability Current Density

The probability current density is a vector quantity that describes the flow of probability in space. It is defined in terms of the wave function and its gradient, and it captures how probability moves from one region to another, ensuring the conservation of probability in quantum dynamics.

Probability Density

In quantum mechanics, the probability density is defined as the product of a wave function and its complex conjugate. It represents the likelihood of finding a particle at a particular position at a given time, and its time evolution is critical for predicting experimental outcomes.

Time Evolution

Time evolution in quantum mechanics is governed by the time-dependent Schrödinger equation. For systems with time-independent Hamiltonians, the time evolution of each energy eigenstate is given by a phase factor exp(-iEt/?), and a general state, being a superposition of eigenstates, evolves by having each term multiplied by its corresponding phase factor.

Continuity Equation in Quantum Mechanics

The continuity equation expresses the conservation of probability, relating the time rate of change of the probability density to the divergence of the probability current density. Mathematically, it is given by ??/?t + ?·J = 0, ensuring that the total probability remains constant over time in an isolated quantum system.

Energy Eigenstates

Energy eigenstates are solutions to the time-independent Schrödinger equation corresponding to specific quantized energy levels. In a confined system like an infinite potential well, these eigenstates form a complete basis set that can be used to represent any admissible wave function of the particle by linear combination.

Infinite Potential Well

An infinite potential well is a quantum mechanical model where a particle is confined to a finite region in space by infinitely high potential barriers. The model provides discrete energy levels and corresponding eigenstates due to the boundary conditions that the wave function must vanish at the walls of the well, leading to quantization of energy.

Superposition Principle

The superposition principle states that any valid quantum state can be expressed as a linear combination of energy eigenstates. This principle is fundamental in quantum mechanics, allowing the construction of initial states that are not eigenstates themselves but instead a sum of several eigenstates, each evolving independently over time.

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