Question

A matter wave is represented by the wave function \[ \Psi(x, y, z, t) = A e^{i(4x+3y+5z-10\pi t)}, \] where A is a constant. The unit vector representing the direction of propagation of this matter wave is
 

• A. \( \frac{4}{5\sqrt{2}} \hat{x} + \frac{3}{5\sqrt{2}} \hat{y} + \frac{1}{\sqrt{2}} \hat{z} \)
• B. \( \frac{3}{5\sqrt{2}} \hat{x} + \frac{4}{5\sqrt{2}} \hat{y} + \frac{1}{5\sqrt{2}} \hat{z} \)
• C. \( \frac{1}{5\sqrt{2}} \hat{x} + \frac{3}{5\sqrt{2}} \hat{y} + \frac{1}{\sqrt{2}} \hat{z} \)
• D. \( \frac{1}{5\sqrt{2}} \hat{x} + \frac{4}{5\sqrt{2}} \hat{y} + \frac{3}{5\sqrt{2}} \hat{z} \)

Hint:

The direction of propagation of a matter wave is given by the normalized wave vector, which can be found from the coefficients of the spatial terms in the wave function.

Answer 1

The Correct Option is A

The given wave function is \[ \Psi(x, y, z, t) = A e^{i(4x+3y+5z-10\pi t)}. \] The general form of the wave function for a matter wave is \( \Psi = A e^{i(k_x x + k_y y + k_z z - \omega t)} \), where \( k_x, k_y, k_z \) are the wave vectors in the \(x\), \(y\), and \(z\) directions, and \( \omega \) is the angular frequency. Comparing the exponents of both wave functions, we find: - \( k_x = 4 \),
- \( k_y = 3 \),
- \( k_z = 5 \).
The direction of propagation of the matter wave is given by the unit vector along the wave vector \( \mathbf{k} \), which is normalized as: \[ \hat{k} = \frac{k_x \hat{x} + k_y \hat{y} + k_z \hat{z}}{\sqrt{k_x^2 + k_y^2 + k_z^2}}. \] Substituting the values of \( k_x, k_y, k_z \), we get: \[ \hat{k} = \frac{4 \hat{x} + 3 \hat{y} + 5 \hat{z}}{\sqrt{4^2 + 3^2 + 5^2}} = \frac{4 \hat{x} + 3 \hat{y} + 5 \hat{z}}{\sqrt{16 + 9 + 25}} = \frac{4 \hat{x} + 3 \hat{y} + 5 \hat{z}}{\sqrt{50}} = \frac{4}{5\sqrt{2}} \hat{x} + \frac{3}{5\sqrt{2}} \hat{y} + \frac{1}{\sqrt{2}} \hat{z}. \] Thus, the unit vector representing the direction of propagation is \( \frac{4}{5\sqrt{2}} \hat{x} + \frac{3}{5\sqrt{2}} \hat{y} + \frac{1}{\sqrt{2}} \hat{z} \), which corresponds to option (A).