Question

Exercises \(78-88\) refer to a particle of mass \(m\) described by the wave function $$ \psi(x)=\left\\{\begin{array}{ll} 2 \sqrt{a^{3}} x e^{-a x} & x>0 \\ 0 & x<0 \end{array}\right. $$ What is the probability that the particle would be found hetween \(x=0\) and \(x=1 / a ?\)

Short answer

The probability that the particle would be found between \(x=0\) and \(x=1/a\) is \[2 - 4e^{-2}\]

Step-by-step solution

- Define the boundary conditions

First, determine the range over which you need to find the probability. In this case, it is between \(x = 0\) and \(x = 1/a\). We need to consider only the positive part of the wave function, as the range is within this domain.

- Construct the Probability Function

The probability function is given by the absolute square of the wave function. So, \(\left|\psi(x)\right|^2 = \left|2\sqrt{a^{3}} x e^{-a x}\right|^2 = 4a^{3}x^{2}e^{-2ax}\).

- Integrate the Probability Function

The total probability is given by the integral of the probability function over the range \(x = 0\) to \(x = 1/a\). This integral is \(\int_{0}^{1/a} 4a^{3}x^{2}e^{-2ax}\, dx\). This can be solved using variable substitution, letting \(u = 2ax, du = 2a dx, dx = du / 2a\). The integral now becomes \(\int_{0}^{2} 2au^{2}e^{-u}\, du\).

- Evaluate the Integral

This is an integral of the form \(\int_{0}^{2} u^{2}e^{-u} du\), which can be solved with integration by parts twice. The solution is \[2 - 4e^{-2}\].

Key concepts

Wave Function

In quantum mechanics, the wave function, often denoted by \( \psi(x) \), is a mathematical expression that captures the quantum state of a system, particularly the behavior of particles like electrons. In our case, the wave function for a particle is given as follows:
\[\psi(x)=\left\{\begin{array}{ll}2 \sqrt{a^{3}} x e^{-a x} & x>0 \0 & x<0\end{array}\right.\]Understanding the wave function is pivotal because it contains all the information necessary to predict how the particle behaves in space. However, the wave function itself doesn't give probabilities directly. Instead, it helps calculate them. The square of the wave function, \( |\psi(x)|^2 \), gives what's known as the probability density function.

Probability Density

The probability density is a crucial concept in quantum mechanics. Given by \( |\psi(x)|^2 \), it indicates how likely it is to find a particle at a specific location. In our exercise, the probability density is:
\[|\psi(x)|^2 = 4a^{3}x^{2}e^{-2ax}\]This expression represents the probability density along the x-axis between positive values where the wave function is non-zero. By integrating this probability density over a certain range, we calculate the likelihood of finding the particle within that specific region.
Understanding probability density is a way of bridging the microscopic, probabilistic world of quantum mechanics and our macroscopic view. It allows us to determine although not precisely, but more accurately, where a particle 'might' be.

Integration by Parts

Integration by parts is a mathematical technique used to solve integrals where standard methods are difficult to apply. It is particularly useful in this exercise when integrating the probability density to find the total probability from \( x = 0 \) to \( x = 1/a \).
The formula for integration by parts is:
\[\int u \, dv = uv - \int v \, du\]This method was applied to solve:
\[\int_{0}^{2} u^{2}e^{-u} du\]By choosing appropriate values for \( u \) and \( dv \), you can simplify the integral into a form that's easier to calculate. Often, integration by parts needs to be used multiple times within the same problem to achieve the desired result. In our exercise, it was applied twice to evaluate the integral only to find the final solution \( 2 - 4e^{-2} \). This showcases its utility in dealing with complex exponential functions, especially in physics problems involving decay and growth.

Particle in a Potential

In quantum mechanics, a "particle in a potential" refers to a scenario where particles move under the influence of a potential field. It forms the basis of many quantum mechanics problems, where solving the Schrödinger equation becomes essential.

• The potential could be anything—like a box, a well, or even a field with varying points.
• For simpler systems, like a particle on a line with respect to a potential, we might idealize it as moving in one dimension.
• This leads to a range of possibilities: bound states, where particles are confined, or unbound states, where they can escape.

The exercise uses a wave function within a specific potential setup, indicating that the behavior and likelihood of the particle between \( x = 0 \) and \( x = 1/a \) depends on the given exponential and polynomial factors in the wave function. Analyzing such scenarios helps comprehend how particles behave at the quantum level in different potential landscapes.