Question

Exercises \(78-88\) refer to a particle of mass \(m\) described by the wave function $$ \psi(x)=\left\\{\begin{array}{ll} 2 \sqrt{a^{3}} x e^{-a x} & x>0 \\ 0 & x<0 \end{array}\right. $$ Determine the expectation value of the momentum of the particle. Explain.

Short answer

Solving the integral after substituting the wave function in the expression \(\langle p \rangle = \int_{-\infty}^{\infty}\psi^*(-x) \hat{p} \psi(x)\, dx, \) it is possible to get the expectation value of the momentum, represented by the real part of the integral's output.

Step-by-step solution

Analyze the wave function

Given is a wave function for a particle of mass \(m\), \(\psi(x)=2\sqrt{a^3}xe^{-ax}\) if \(x>0\) and \(\psi(x)=0\) if \(x<0\). So, the particle is described totally in terms of this wave-function when \(x>0\). Probability density for each position of the particle can be found with \(|\psi(x)|^2\). Considering both mathematical and physical meaning of the wave-function can lead to the solution.

Expectation value for momentum operator

The formula to calculate the expectation value for an operator \(\hat{O}\) in quantum mechanics is \(\langle O \rangle = \langle \psi | \hat{O} | \psi \rangle\). For the 1-dimensional quantum mechanical momentum operator, we get \(\hat{p} = -i\hbar \frac{d}{dx}\). Therefore, the desired expectation value is \(\langle p \rangle = \langle \psi | \hat{p} | \psi \rangle.\) Which implies we need to solve the integral \[\int_{-\infty}^{\infty}\psi^*(-x) \hat{p} \psi(x)\, dx=\int_0^{\infty} \psi^*(-x) (-i \hbar \frac{d}{dx}) \psi(x)\, dx.\] Here \(\psi^*(-x)\) represents the complex conjugate of the wave function.

Solving the integral

Proceed by substituting \(\psi(x)\) and \(\psi^*(-x)\) in the integral and simplifying it. After solving the integral, real part represents the expectation value for momentum.

Key concepts

Momentum Operator

The momentum operator in quantum mechanics is a critical concept that plays a vital role in determining the behavior of particles at a quantum scale. In the context of one dimension, the momentum operator is denoted by \( \hat{p} \) and is mathematically represented by \( -i\hbar\frac{d}{dx} \) where \( i \) is the imaginary unit, \( \hbar \) (pronounced 'h-bar') is the reduced Planck’s constant, and \(\frac{d}{dx}\) signifies the derivative with respect to position \( x \).

Applying this operator to a wave function \( \psi(x) \) within an integral allows us to calculate the expected, or averaged, value of momentum for a quantum state. It's pivotal to grasp the momentum operator because it links the wave-like nature of particles to their momentum, a classical concept, through the mathematical process of differentiation.

Wave Function

A wave function \( \psi(x) \) in quantum mechanics is a mathematical description of the quantum state of a system. Its absolute square, \( |\psi(x)|^2 \) indicates the probability density of finding a particle at a particular location \( x \). Because particles at a quantum level display both wave-like and particle-like properties, the wave function encompasses this duality and provides probabilities rather than definite predictions.

The given wave function for a particle \( \psi(x)=2\sqrt{a^3}xe^{-ax} \) for \( x>0 \) and \( \psi(x)=0 \) for \( x<0 \) is a piecewise function that is specific to the particle's position and is key to computing quantities like momentum and energy.

Probability Density

Probability density in quantum mechanics corresponds to the likelihood of finding a particle in a given region of space. Defined by the squared modulus of the wave function \( |\psi(x)|^2 \) for a continuous variable \( x \) in one dimension, it ensures that the total probability of finding the particle somewhere in space is 1. Integrating probability density over all space gives a normalized result, clarifying that the particle must exist somewhere within the specified bounds.

For the prescribed wave function, probability density is not uniform and varies depending on position \( x \), since it depends on both the shape of the wave function and the exponential decay term present in \( \psi(x) \).

Complex Conjugate

The complex conjugate of a wave function is an essential concept when dealing with quantum states that may have complex values. It refers to changing the sign of the imaginary part of the complex numbers present in the wave function. Symbolically, if \( \psi(x) \) is a wave function, then its complex conjugate is denoted as \( \psi^*(x) \).

In quantum mechanical operations, especially those involving inner products or expectation values, we often have to multiply the wave function by its complex conjugate to ensure a real number outcome, such as in probability density calculations or when finding expectation values, as is the case in our exercise.

Integration in Quantum Mechanics

Integration is a key mathematical process in quantum mechanics, used extensively in the normalization of wave functions and the calculation of expectation values. In the context of finding the expectation value of momentum, integration involves evaluating an integral that typically extends over all space (from \( -\infty \) to \( +\infty \) for one dimensional cases).

The integrals often require the product of the wave function, its complex conjugate, and the operator representing the observable of interest. Such integrals can be quite complex and typically require techniques from calculus to solve them. In our exercise, the integral simplifies due to the wave function being zero for \( x<0 \) and becomes an integral from \(0\) to \(\infty\) encompassing the non-zero region of the wave function.