Try the fastest way to create flashcards
Question

# Consider a freely moving quantum particle with mass m and speed v. Its energy is $E=K+U=\frac{1}{2} m v^{2}+0$. Determine the phase speed of the quantum wave representing the particle and show that it is different from the speed at which the particle transports mass and energy.

Solution

Verified
Step 1
1 of 2

As we know, the particle has energy :

\begin{align*} E&=K+U\\ E&=\frac{1}{2}mv^2+0\\ E&=K=\frac{1}{2}mv^2\\ E&=hf \\ \Rightarrow \lambda&=\frac{h}{mv} \tag{Wavelength.}\\ \Rightarrow v_{p}&=f \lambda \tag{Phase velocity \rightarrow equation 1.}\\ \end{align*}

Now, we have to substitute values in equation 1 and solve this equation :

\begin{align*} v_{p}&=f \lambda \tag{Equation 1.}\\ v_{p}&=\frac{E}{h} \cdot \frac{h}{mv} \\ v_{p}&=\frac{\frac{1}{2} mv^2}{h} \cdot \frac{h}{mv} \tag{Substitute values in equation.}\\ v_{p}&=\frac{mv^2}{2h} \cdot \frac{h}{mv} \\ v_{p}&=\frac{v}{2} \ne v \\\ \end{align*}

## Recommended textbook solutions #### Physics for Scientists and Engineers: A Strategic Approach with Modern Physics

4th EditionISBN: 9780133942651 (8 more)Randall D. Knight
3,508 solutions #### Mathematical Methods in the Physical Sciences

3rd EditionISBN: 9780471198260 (1 more)Mary L. Boas
3,355 solutions #### Modern Physics for Scientists and Engineers

3rd EditionISBN: 9780534493394Clement J. Moses, Curt A. Moyer, Raymond A. Serway
747 solutions #### Fundamentals of Physics

10th EditionISBN: 9781118230718 (3 more)David Halliday, Jearl Walker, Robert Resnick
8,971 solutions