Question

A free particle moving in one dimension has wave function $$\Psi(x, t) = A[e^{i(kx-vt)} - e^{i(2kx-4vt)}]$$ where \(k\) and \(\omega\) are positive real constants. (a) At \(t\) = 0 what are the two smallest positive values of \(x\) for which the probability function \(\mid \Psi(x, t) \mid ^2\) is a maximum? (b) Repeat part (a) for time \(t = 2\pi/\omega\). (c) Calculate \(v_{av}\) as the distance the maxima have moved divided by the elapsed time. Compare your result to the expression \(v_{av} = (\omega_2 - \omega_1)/(k_2 - k_1)\) from Example 40.1.

Short answer

(a) \(x = \frac{\pi}{k}\), \(x = \frac{3\pi}{k}\); (b) Same \(x\) values; (c) \(v_{av} = \frac{\omega}{k}\).

Step-by-step solution

Simplify the Wave Function at t = 0

Set time \( t = 0 \) in the wave function: \[\Psi(x, 0) = A \left[ e^{ikx} - e^{i2kx} \right] = A (e^{ikx} - e^{i2kx}).\]This expression is a linear combination of complex exponentials, which can be simplified using Euler’s formula \( e^{ix} = \cos(x) + i\sin(x) \). The probability density function is then expressed as:\[|\Psi(x,0)|^2 = |A|^2 |e^{ikx} - e^{i2kx}|^2.\]

Find the Probability Density Function at t = 0

To obtain the probability density function, compute the modulus squared:\[|\Psi(x, 0)|^2 = |A|^2 (e^{ikx} - e^{i2kx})(e^{-ikx} - e^{-i2kx}).\]Expanding, this gives:\[|\Psi(x, 0)|^2 = |A|^2 (2 - e^{ikx} - e^{-ikx}) = 2|A|^2(1 - \cos(kx)).\]

Determine x for Maximum Probability at t = 0

The probability density function \(|\Psi|^2\) is maximized when \(\cos(kx) = -1\). This occurs at:\[kx = (2n + 1)\pi,\]where \(n\) is an integer. Solving for \(x\):\[x = \frac{(2n+1)\pi}{k}.\]The two smallest positive values for \(x\) are \(x = \frac{\pi}{k}\) and \(x = \frac{3\pi}{k}\).

Simplify the Wave Function at t = 2\pi/\omega

For \(t = \frac{2\pi}{\omega}\), substitute into the wave function:\[\Psi(x, \frac{2\pi}{\omega}) = A \left[ e^{i(kx- \frac{2\pi}{\omega}v)} - e^{i(2kx - 4\frac{2\pi}{\omega}v)} \right].\]Since \(v = \omega/k\), this simplifies to:\[\Psi(x, \frac{2\pi}{\omega}) = A \left[ e^{ikx - i2\pi} - e^{i2kx - i8\pi} \right].\] Since exponentials of multiples of \(-2\pi i\) are equal to 1, this yields:\[\Psi(x, \frac{2\pi}{\omega}) = A(e^{ikx} - e^{i2kx}).\]

Find the Probability Density Function at t = 2\pi/\omega

Evaluate the modulus squared as at \(t = 0\):\[|\Psi(x, \frac{2\pi}{\omega})|^2 = 2|A|^2(1 - \cos(kx)).\]This is identical to the expression at \(t = 0\), so the maxima occur at the same values of \(x\).

Calculate Average Velocity of Maximum Positions

The maxima move from \(x_1 = \frac{\pi}{k}\) to \(x_2 = \frac{3\pi}{k}\) over time \(t = \frac{2\pi}{\omega}\). Calculate the distance moved:\[\Delta x = \frac{3\pi}{k} - \frac{\pi}{k} = \frac{2\pi}{k}.\]The average velocity \(v_{av}\) is given by:\[v_{av} = \frac{\Delta x}{\Delta t} = \frac{\frac{2\pi}{k}}{\frac{2\pi}{\omega}} = \frac{\omega}{k}.\]Compare this with \(v_{av} = \frac{\omega_2 - \omega_1}{k_2 - k_1}\), which for this problem also results in \(\frac{\omega}{k}\).

Key concepts

Wave Function

Wave functions are fundamental in quantum mechanics, offering a mathematical depiction of a quantum system. They encapsulate the system's state and information about a particle's position and momentum. For a free particle moving in one dimension, the wave function is often expressed as a combination of complex exponentials. In our problem, the wave function \[\Psi(x, t) = A[e^{i(kx-vt)} - e^{i(2kx-4vt)}]\] is used.Understanding this involves a few key points:- **Complex Exponentials**: These are parts of the wave function that describe oscillatory behavior, ensuring we can capture properties such as frequency and wavelength.- **Linear Combinations**: The function is a blend or superposition of two exponential terms, which is critical in quantum mechanics because it reflects the principle of superposition, allowing particles to exist in multiple states simultaneously.Ultimately, the wave function allows us to compute probabilities of finding particles in particular states, crucial for solving quantum mechanics problems.

Probability Density

Probability density functions are derived from wave functions and give the likelihood of locating a particle at a specific position. For a wave function \( \Psi(x, t) \), the probability density is given by the modulus squared, \( |\Psi(x, t)|^2 \).In the exercise, at time \( t = 0 \) and \( t = \frac{2\pi}{\omega} \), the resulting probability density is:\[ |\Psi(x, t)|^2 = 2|A|^2(1 - \cos(kx)). \]Here are a few important aspects:- **Maximizing Probability**: The density is maximized when \( \cos(kx) = -1 \), leading to specific quantized locations \( x \) where the probability is highest.- **Physical Meaning**: These maxima correspond to where a particle is most likely to be found. Calculating these points is essential for predicting particle behavior.This reflects the probabilistic nature of quantum mechanics, unlike classical physics, where predictions are deterministic.

Average Velocity

The concept of average velocity in quantum mechanics for probability maxima provides insight into how wave peaks move over time. In this problem, we determine the average velocity \( v_{av} \) using the relationship:\[ v_{av} = \frac{\Delta x}{\Delta t}. \]A few key points about this are:- **Maxima Movement**: By noting the shift from \( x_1 = \frac{\pi}{k} \) to \( x_2 = \frac{3\pi}{k} \) over a time period \( \Delta t = \frac{2\pi}{\omega} \), we calculate how fast the maxima move.- **Time and Spacetime Averaging**: These calculations average position changes over specified time intervals, echoing concepts of velocity known from classical physics but adapted to quantum wave dynamics.The calculated average velocity aligns with hypothetical velocity expressions from theoretical examples, reinforcing the mathematical consistency of quantum mechanics.

Complex Exponentials

Complex exponentials are fundamental in quantum physics, often emerging from wave function solutions. These functions, expressed in Euler's form \( e^{ix} = \cos(x) + i\sin(x) \), define oscillating phenomena key to understanding wave mechanics.When applied, complex exponentials:- **Enable Representation**: They allow wave functions to efficiently represent periodic behaviors in quantum systems.

• **Concisely encode information**: Amplitude and phase are represented compactly.
• **Facilitate Calculations**: Operations like differentiation and integration are simplified.

- **Interference and Superposition**: They underpin these key quantum mechanics concepts, essential for describing how waves interact through constructive and destructive interference.Thus, they play a critical role in both mathematical formulations and in elucidating physical phenomena involving waves and particles in quantum mechanics.