Question

An electron is moving as a free particle in the -$x$-direction with momentum that has magnitude 4.50 $\times$ 10${}^{-24}$ kg $\bullet$ m/s. What is the one- dimensional time-dependent wave function of the electron?

Short answer

$\mathrm{\Psi }(x,t)=A{e}^{i((4.268\times {10}^{10})x-(1.052\times {10}^{17})t)}$.

Step-by-step solution

Identify the Given Values

We are given that the electron has a momentum, $p$, with a magnitude of $4.50\times {10}^{-24}$ kg·m/s, and it is moving in the $-x$-direction.

Write down the Expression for the Wave Function

The time-dependent wave function for a free particle can be expressed as: $$\mathrm{\Psi }(x,t)=A{e}^{i(kx-\omega t)},$$where $A$ is the amplitude, $k$ is the wave number, and $\omega$ is the angular frequency.

Determine the Wave Number $k$

The wave number $k$ is related to the momentum $p$ by the equation:$$k=\frac{p}{\hslash },$$where $\hslash$ is the reduced Planck's constant, $\hslash =1.0545718\times {10}^{-34}$ J·s.Substitute $p=4.50\times {10}^{-24}$ kg·m/s:$$k=\frac{4.50\times {10}^{-24}}{1.0545718\times {10}^{-34}}=4.268\times {10}^{10}{\text{ m}}^{-1}.$$

Determine the Angular Frequency $\omega$

The angular frequency $\omega$ is related to the energy $E$, and since $E=\frac{p^2}{2m}$, we can find $\omega$ using:$$\omega =\frac{E}{\hslash }.$$Assume the rest mass of the electron is $m=9.10938356\times {10}^{-31}$ kg:First compute $E$:$$E=\frac{(4.50\times {10}^{-24}{)}^2}{2\times 9.10938356\times {10}^{-31}}=1.11\times {10}^{-17}\text{ J}.$$Then compute:$$\omega =\frac{1.11\times {10}^{-17}}{1.0545718\times {10}^{-34}}=1.052\times {10}^{17}{\text{ s}}^{-1}.$$

Construct the Wave Function

Substituting $k$ and $\omega$ into the wave function equation:$$\mathrm{\Psi }(x,t)=A{e}^{i((4.268\times {10}^{10})x-(1.052\times {10}^{17})t)}.$$This represents the time-dependent wave function of the electron.

Key concepts

Wave Function

In quantum mechanics, the wave function forms the cornerstone that describes the quantum state of a particle. It is a complex-valued function, often denoted by the Greek letter psi ($\mathrm{\Psi }$). For a free particle, like an electron, the wave function varies with both position $x$ and time $t$.

The general form of a one-dimensional time-dependent wave function is expressed as:

• $\mathrm{\Psi }(x,t)=A{e}^{i(kx-\omega t)}$

Here, $A$ represents the amplitude, which dictates the probability density of finding the particle at a particular position and time. Although the wave function itself may not hold direct physical meaning, its absolute square, $|\mathrm{\Psi }(x,t){|}^2$, gives the probability density.

The wave function encapsulates the duality of particles behaving like waves, explaining interference and diffraction phenomena central to quantum mechanics.

Momentum

Momentum in quantum mechanics is a fundamental property related to the motion of particles, similar to classical physics. However, the quantum description incorporates both wave and particle nature.

For an electron described by a wave function, its momentum $p$ is tied to the wave number $k$ via the reduced Planck's constant $\hslash$:

• $k=\frac{p}{\hslash }$

This relationship shows how momentum is interconnected with the wave-like traits of the particle. It highlights quantum mechanics' departure from purely classical concepts, by quantifying the necessity of its wave behavior.

For a moving electron, with a given momentum magnitude $p=4.50\times {10}^{-24}$ kg·m/s, the wave character and its corresponding properties can thus be quantified through calculations like these. This blend of momentum with quantum behavior aids in predicting the particle's behavior and underlying probabilities.

Angular Frequency

Angular frequency $\omega$ links closely with the energy of the quantum particle. Similarly, as traditional frequency connects to cycles per time, angular frequency deals with radians per second.

In the context of our wave function, it reflects how fast the wave oscillates in time. Specifically:

• $\omega =\frac{E}{\hslash }$

Here, $E$ stands for the particle's energy, which is acquired from its momentum and mass: $E=\frac{p^2}{2m}$. This formula relates the energy of the system back to its quantum nature through $\hslash$, providing insight on temporal variations seen in the wave's progression.

For an electron moving with our given momentum and known mass, the angular frequency can be precisely calculated. This constitutes an essential part of understanding how quantum particles evolve over time and interact with their environment.

Wave Number

Wave number $k$ is a concept depicting the spatial frequency of a wave, equivalent to the number of wave cycles per unit distance. In quantum mechanics, it translates the momentum of particles into wave-like characteristics.

It’s defined as:

• $k=\frac{p}{\hslash }$

Thus, $k$ offers an intrinsic link between position and momentum - pivotal to the uncertainty principle at the heart of quantum mechanics. Higher $k$ values suggest tightly packed wave fronts, and thus more rapid spatial oscillations.

For our electron example, with a given momentum, determining $k$ allows us to express the wave function accurately in space. Understanding $k$ offers a deeper insight into how a quantum particle distributes its presence across a spatial domain and sets groundwork for solving many quantum mechanical problems.