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}^{-ax}& x>0\\ 0& x<0\end{array}$$\right. $$ Calculate the uncertainty in the particle's momentum.

Short answer

The uncertainty in the particle's momentum can be calculated by finding the difference between mean square momentum and the square of the mean momentum using the wavefunction. The mathematical process involves evaluating certain integrals and differentiating the wavefunction. The exact numerical result depends on the concrete normalization constant $a$ and mass $m$ of a particle.

Step-by-step solution

Mean value of square of momentum

The operator for the square of the momentum is $p^2=-{\hslash }^2\frac{d^2}{d{x}^2}$. So, the mean value is given by the following integral: $$⟨p^2⟩=-{\hslash }^2{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\psi }^{\ast }(x)\frac{d^2}{d{x}^2}\psi (x)dx$$ Also, consider that ${\psi }^{\ast }(x)=\psi (x)$ because the wavefunction is real.

Square of the mean value of momentum

The operator for the momentum is $p=-i\hslash \frac{d}{dx}$. So, the square of the mean value of the momentum is given by: $$⟨p{⟩}^2={(-i\hslash {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\psi }^{\ast }(x)\frac{d}{dx}\psi (x)dx)}^2$$

Use the uncertainty relation

The uncertainty of the momentum is given by the following equation according to Heisenberg's uncertainty principle: $$(\mathrm{\Delta }p{)}^2=⟨p^2⟩-⟨p{⟩}^2$$ Substitute the result from step 1 and step 2 in this equation to get the desired result.

Key concepts

Uncertainty Principle

The Heisenberg Uncertainty Principle is a key underpinning of quantum mechanics, highlighting the intrinsic limitations in measuring complementary properties of a particle, such as position and momentum. It states that the more precisely one property is known, the less precisely the other can be known, encapsulated by the formula:
$$\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{\hslash }{2}$$where $\mathrm{\Delta }x$ is the uncertainty in position and $\mathrm{\Delta }p$ is the uncertainty in momentum, and $\hslash$ is the reduced Planck’s constant.
This principle emphasizes the probabilistic nature of quantum mechanics, reflecting the idea that particles do not have definite locations or speeds at the atomic scale until they are measured. Instead, these properties are described by probabilities.

Wave Function

A wave function, commonly denoted by $\psi (x)$, is a mathematical description of the quantum state of a particle or system. It contains all the necessary information about a system's state and its interaction with the environment. In quantum mechanics, the wave function is a solution to the Schrödinger equation.
The square of its absolute value, $|\psi (x){|}^2$, gives the probability density of finding a particle at a specific position. For example, in the given function:
$$\psi (x)=\{\begin{array}{lll}2\sqrt{a^3}x{e}^{-ax}& x>00& x<0\end{array}$$it shows us that the probability of finding the particle decreases exponentially as $x$ increases for $x>0$, and it is zero for $x<0$.
This function shapes the way we predict a particle's behavior within a quantum system.

Momentum Operator

In quantum mechanics, the momentum operator is a crucial tool that represents the momentum of a particle in the position space representation. It is given by the expression:
$$\hat{p}=-i\hslash \frac{d}{dx}$$where $\hat{p}$ is the momentum operator, $\hslash$ is the reduced Planck's constant, and $i$ is the imaginary unit.
This operator functions as part of the Schrödinger equation, acting on wave functions to yield momentum-related information.

• In the given exercise, the momentum operator is applied to the wave function $\psi (x)$ to determine the mean momentum.
• The momentum operator allows us to calculate expected values when used with the wave functions, highlighting the deep connections between states and measurements in quantum systems.

By working with this operator, physicists can predict how particles behave, offering insights into their dynamic properties.