Show that the free-particle one-dimensional Schrödinger wave equation (6.8) is invariant with respect to galilean transformations. Do this by showing that, when the transformation \(x^{\prime}=x-v t, t^{\prime}=t\) is applied, the transformed wave function \(\psi^{\prime}\left(x^{\prime}, t^{\prime}\right)=f(x, t) \psi(x, t)\) satisfies Eq. (6.8) with respect to the primed variables, where \(f\) involves only \(x, t, \hbar, m\), and \(v .\) Find the form of \(f\), and show that the traveling wave solution \(\psi(x, t)=A e^{i(k x-\omega t)}\) transforms as expected.

The Schrödinger wave equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. For a free particle in one dimension, the equation is given by:\[ i \frac{\text{∂}\tilde{\tilde{ψ}}} {\text{∂}t} = - \frac{\tilde{\tilde{h}}^2}{2m} \frac{\text{∂}\tilde{\tilde{2}}\tilde{\tilde{ψ}}}{\text{∂}x^2} \]In this equation:

• \( i \) is the imaginary unit.
• \( \tilde{\tilde{ψ}} \) is the wave function which describes the state of the particle.
• \( \tilde{\tilde{h}} \) stands for Planck's constant divided by \( 2 \tilde{\tilde{π}} \).
• \( m \) is the mass of the particle.
• \( t \) is time and \( x \) is the spatial coordinate.

The wave function \( \tilde{\tilde{ψ}} (x, t) \) contains all the information about a quantum system, and its squared magnitude \( | \tilde{\tilde{ψ}} (x, t) | ^2 \) represents the probability density of finding a particle at position \( x \) and time \( t \).Understanding the Schrödinger equation is crucial as it lays the foundation for quantum mechanics, allowing us to analyze how particles behave at a quantum level.

In the context of quantum mechanics, wave function transformations allow us to explore how wave functions change under various conditions, such as shifting frames of reference. For this exercise, we are particularly interested in Galilean transformations, which deal with non-relativistic motion.A Galilean transformation in 1D can be expressed as:\( x' = x - vt \) and \( t' = t \)Where \( x' \) is the new position coordinate in a frame moving with velocity \( v \) and \( t \) remains the same. Now, the wave function transformation given in this problem is:\( ψ'(x', t') = f(x, t) ψ(x, t) \)Here, \( f(x, t) \) is a function that depends on position, time, the reduced Planck's constant \( \hbar \), mass \( m \), and the velocity \( v \). To maintain the Schrödinger equation form under transformation, we need to find an appropriate form of \( f(x, t) \). Through work, we found that:\( f(x, t) = e^{i \frac{m v x}{ \tilde{\tilde{h}} } } e^{-i \frac{m v^2 t}{2 \tilde{\tilde{h}} }} \)This form cancels out extra terms introduced due to the Galilean shift, thus preserving the integrity of the Schrödinger equation in the transformed frame. This analysis shows how wave functions adapt under changes in the observer's frame, a key insight when dealing with moving systems.

A traveling wave solution is a specific form of wave function used to describe free particles moving through space. In this context, the wave function is given by:\( ψ(x, t) = A e^{i(kx - \tilde{\tilde{ω}} t)} \)Where:

• \( A \) is the amplitude of the wave.
• \( k \) is the wave number, relating to the wavelength \( λ \) by \( k = \frac{2 \tilde{\tilde{π}}}{λ} \).
• \( \tilde{\tilde{ω}} \) is the angular frequency.

This solution shows how the wave function of a particle moves along the \( x \)-axis over time.Now, when the Galilean transformation \( x' = x - vt \) is applied, we can substitute to find the new wave function as:\( ψ'(x', t) = A e^{i[k(x' + vt) - \tilde{\tilde{ω}} t]} \)Expanding and using the function \( f(x, t) \) as derived earlier, we get:\( ψ'(x', t) = A e^{i(k x' + (kv - \tilde{\tilde{ω}}) t)} \)This shows that the traveling wave behaves predictably under a Galilean transformation, maintaining its structure while shifting in space and time according to the frame's velocity. This consistency is crucial for confirming that the Schrödinger equation and its solutions abide by fundamental physical principles.