Problem 13 - Classical Dynamics of Particles and Systems

Question

Consider the simplified wave function $$ \Psi(x, t)=A e^{i(\omega t-k x)} $$ Assume that \(\omega\) and \(v\) are complex quantities and that \(k\) is real: $$ \begin{array}{l} \omega=\alpha+i \beta \\ v=u+i w \end{array} $$ Show that the wave is damped in time. Use the fact that \(k^{2}=\omega^{2} / v^{2}\) to obtain ex. pressions for \(\alpha\) and \(\beta\) in terms of \(u\) and \(w .\) Find the phase velocity for this case.

Step-by-Step Solution

Answer

We rewrite the wave function \(\Psi(x, t)\) considering the complex quantities of \(\omega\) and \(v\), and show that the damping in time is represented by \(e^{-\beta t}\). By using the relation \(k^2 = \frac{\omega^2}{v^2}\) and equating real and imaginary parts, we find expressions for \(\alpha\) and \(\beta\) in terms of \(u\) and \(w\): \[ \alpha = u\frac{k}{\sqrt{u^2 - w^2}}; \quad \beta = w\frac{k}{\sqrt{u^2 - w^2}} \] The phase velocity is determined as: \[ v_p = \frac{u + iw}{\sqrt{u^2 - w^2}} \] Thus, we successfully show that the wave is damped in time and find the expressions for \(\alpha\), \(\beta\), and the phase velocity.

Step 1: 1. Rewrite the wave function with real and imaginary parts

First, let's consider \(\omega\) and \(v\) as complex quantities as mentioned in the exercise, and rewrite the wave function \(\Psi(x, t)\) using this information: \[ \Psi(x, t) = Ae^{i[\alpha + i\beta]t - ikx} = A e^{i(\alpha t - kx)}e^{-\beta t} \]

Step 2: 2. Separate real and imaginary components

Now, let's use Euler's formula to separate the real and imaginary components of the wave function: \[ \Psi(x, t) = A \left[\cos{(\alpha t - kx)} - i \sin{(\alpha t - kx)}\right] e^{-\beta t} \]

Step 3: 3. Show that the wave is damped in time

Observe that the wave function is multiplied by the exponent \(e^{-\beta t}\) and since \(\beta\) is real and positive, its exponential component will decay with time. This behavior of the exponential shows the damping of the wave in time.

Step 4: 4. Obtain expressions for \(\alpha\) and \(\beta\) in terms of \(u\) and \(w\)

Now we will use the given relation \(k^2 = \frac{\omega^2}{v^2}\) and substitute the complex values of \(\omega\) and \(v\): \[ k^2 = \frac{(\alpha + i\beta)^2}{(u + iw)^2} \] First, let's square both \(\omega\) and \(v\): \(\omega^2 = (\alpha + i\beta)(\alpha + i\beta) = \alpha^2 + 2i\alpha\beta - \beta^2\\ v^2 = (u + iw)(u + iw) = u^2 - w^2 +2i uw\) Substitute these values in the given relation and equate real and imaginary parts as separate equations: Real part: \[ k^2 = \frac{\alpha^2 - \beta^2}{u^2 - w^2} \] Imaginary part: \[ 0 = \frac{2\alpha\beta}{2uw} \] From the imaginary part, we get \(\alpha\beta = 0\). Since \(\alpha\) and \(\beta\) are nonzero, we can derive expressions for them in terms of \(u\) and \(w\): \[ \alpha = u\frac{k}{\sqrt{u^2 - w^2}}; \quad \beta = w\frac{k}{\sqrt{u^2 - w^2}} \]

Step 5: 5. Find the phase velocity

To find the phase velocity, we use the formula \(v_p = \frac{\omega}{k}\). Considering that \(\omega = \alpha + i\beta\), we get: \[ v_p = \frac{\alpha + i\beta}{k} = \frac{u\frac{k}{\sqrt{u^2 - w^2}} + i w\frac{k}{\sqrt{u^2 - w^2}}}{k} = \frac{u + iw}{\sqrt{u^2 - w^2}} \] By substituting the expressions for \(\alpha\), and \(\beta\), we obtain the phase velocity in terms of \(u\), \(w\), and \(v\). In this complex form, the real part corresponds to the phase velocity in the x-direction, while the imaginary part represents the damping/attenuation per wavelength.

Key Concepts

Complex Wave Function Analysis

When studying wave functions, we sometimes encounter complex wave functions that can be used to describe phenomena such as damped oscillations or waves traveling through a medium that absorbs energy. The given wave function,
\[\Psi(x, t)=A e^{i(\omega t-k x)}\]
where \(\omega\) and \(v\) are complex numbers, encapsulates both the oscillatory nature of waves and the damping effects over time.

To analyze this complex wave function, we use the knowledge that complex numbers can be expressed in terms of their real and imaginary parts. Euler's formula, \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\), is crucial here as it helps us link the exponential function with sine and cosine components.

In the given solution, after separating the complex exponentials' real and imaginary parts, the exponential decay factor, \(e^{-\beta t}\), becomes evident. This factor indicates that as time increases, the amplitude of the wave decreases, showing us a clear physical interpretation of damping within the wave's behavior. By understanding complex wave function analysis, students can comprehend how waves can exhibit such diminishing amplitude without having to look at the energy dissipation mechanisms in detail.

Exponential Decay in Waves

Exponential decay is a term that describes how certain quantities decrease at a rate proportional to their current value. For wave functions, it's a way to represent how the amplitude of a wave diminishes over time due to some form of resistance or friction in the medium through which the wave is traveling.

In our exercise, the presence of the term \(e^{-\beta t}\) in the wave function indicates that the wave amplitude is subject to exponential decay. Here, \(\beta\) is a positive real number that represents the rate of decay. The importance of understanding exponential decay in waves lies in its widespread applicability, from quantum mechanics to electrical engineering.

Exponential decay can often be attributed to various factors, such as the medium's absorptivity, resistive forces, or even a loss of energy due to radiation. In practical terms, this means that the wave's energy is not conserved but instead dissipates over distance or time. Recognizing the signature of exponential decay in a wave function allows students to predict how quickly a wave will lose its strength as it propagates.

Phase Velocity

Phase velocity is a fundamental concept in wave mechanics that describes the speed at which a given phase of the wave travels through space. It is defined as the ratio of the angular frequency \(\omega\) to the wave number \(k\), denoted by \(v_p = \frac{\omega}{k}\). This ratio gives us the velocity of a specific phase point, such as the crest of a wave, as it moves through the medium.

In our solution, we find that when \(\omega\) is a complex quantity, we end up with a complex expression for phase velocity. It indicates not just the speed of phase propagation but also includes information about the damping of the wave. The real part of the phase velocity pertains to the actual speed of the wave phases moving in space, while the imaginary part relates to the attenuation of the wave.The concept of phase velocity is vital in designing systems that rely on wave propagation, such as communication systems, where signals must be transmitted over significant distances without losing their integrity. Understanding the relationship between the phase velocity, wave frequency, and medium properties allows students to analyze and predict the behavior of waves in various contexts.