Question

Exercises \(94-97\) refer to a bound particle of mass \(m\) described by the wave function $$ \psi(x)=A x e^{-x^{2} / 2 b^{2}} $$ Given that the particle's total energy \(E\) is 0 , show that the potential energy is \(U(x)=\frac{h^{2}}{2 m b^{4}} x^{2}-\frac{3 h^{2}}{2 m b^{2}}\)

Short answer

The potential energy of the particle is given by \(U(x)=\frac{h^{2}}{2 m b^{4}} x^{2}-\frac{3 h^{2}}{2 m b^{2}}\)

Step-by-step solution

Compute the first derivative of the wave function

The wave function \(\psi(x)\) is given as \(A x e^{-x^{2} / 2 b^{2}}\). We need to compute its first derivative. Using the product and chain rule, it is:\(\psi'(x) = Ae^{-x^2/2b^2} - Ax^2/b^2 e^{-x^2/2b^2}\)

Compute the second derivative of the wave function

We compute the second derivative of the wave function from the result in Step 1. This also involves application of product and chain rule:\(\psi''(x) = -Ax^2/b^2 e^{-x^2/{2b^2}} - A/b^2 e^{-x^2/{2b^2}} + 2Ax^3/{b^4} e^{-x^2/{2b^2}}\)

Calculate the kinetic energy

From the Schrödinger equation, the kinetic energy is given by \(-\frac{h^2}{2m}\psi''(x)\). Substituting \(\psi''(x)\) with the result from Step 2 gives:\(K(x) = -\frac{h^2}{2m}[-Ax^2/b^2 e^{-x^2/{2b^2}}- A/b^2 e^{-x^2/{2b^2}}+ 2Ax^3/b^4 e^{-x^2/{2b^2}}]\) = \[\frac{h^2x^2}{2mb^2} - \frac{h^2}{2mb^2} + \frac{h^2x^3}{mb^4}\] \(\psi(x)\)

Find the expression for the potential energy

Knowing that the total energy is zero, we can express the potential energy as \(U=E - K\). Hence \(U(x) = - K(x)\). Substitute \(K(x)\) from Step 3 to get:\(U(x) = -\[\frac{h^2x^2}{2mb^2} - \frac{h^2}{2mb^2} + \frac{h^2x^3}{mb^4}\] \(\psi(x)\) = \frac{h^2x^2}{2mb^2} - \frac{3h^2}{2mb^2}\)

Key concepts

Wave Function

In the realm of quantum mechanics, the wave function is a fundamental concept. It represents the quantum state of a particle in space and time. The wave function, usually denoted by the Greek letter \( \psi(x) \), is a crucial part of describing the behavior of particles at a subatomic level.

• It's a complex-valued function of position \( x \) and possibly time.
• The square of the wave function's absolute value, \(|\psi(x)|^2\), gives us the probability density of finding a particle at a particular position \( x \).

In the given exercise, the wave function is provided as \( \psi(x)=A x e^{-x^2 / 2b^2} \). This indicates a Gaussian-type function, which illustrates how the particle's probability amplitude distributes around a center point.

Understanding this is essential because at the quantum level, particles don't possess definite positions. Instead, they are defined by probabilities, which the wave function helps calculate.

Potential Energy

Potential energy in quantum mechanics is a measure of the potential energy landscape that affects a particle's dynamics. It is integral in understanding how particles behave under different forces and configurations.

• In classical physics, potential energy is often associated with forces like gravity or electromagnetism.
• In quantum mechanics, it forms a core component of the total energy of a system, often represented as \( U(x) \).

The exercise derives the potential energy using the expression \( U(x) = \frac{h^2}{2mb^4}x^2 - \frac{3h^2}{2mb^2} \). This formula shows a potential that changes with \( x \), indicating variable influence on particle behavior as its position changes.

The expression reflects the system's potential energy where the total energy is set to zero, aligning with the principle that waves can oscillate around potential energy wells. Grasping how potential energy shapes the wave function is key to mastering quantum mechanics.

Schrödinger Equation

The Schrödinger Equation is a cornerstone of quantum mechanics. It allows us to calculate how quantum systems evolve over time and describes how a particle's wave function changes.

• The equation is given as \( -\frac{h^2}{2m} \frac{d^2\psi}{dx^2} + U(x)\psi = E\psi \).
• This equation combines both kinetic and potential energy terms into one framework.

For the given exercise, we use the time-independent Schrödinger Equation. Here, the total energy \( E \) is a constant, set to zero. It is used to equate potential and kinetic energy in the expression \( U = E - K \), as done in the step-by-step solution.

Understanding and applying the Schrödinger Equation lets us predict how quantum states progress, making it pivotal for anyone delving into quantum behavior. It integrates the wave function, potential energy, and total energy, highlighting their interconnections.