PX156 - C2 - matter waves

• planck:

• de broglie:

• $\psi =\cos (kx-\omega t)$ is not a solution of TDSE

normalization condition

• max born proposed:

which can be used to calculate the probability of finding the particle between $a$ and $b$, at time $t$

• the normalization condition:

- the particle should be somewhere in all of space
- ie: $|\psi {|}^2\to 0$ as $x\to \pm \mathrm{\infty }$

• plane wave:

• therefore plane waves cannot represent matter waves

• solution: wave packet

  • ${\psi }_1$ and ${\psi }_2$ are the solutions of the TDSE:
  $$-\frac{{\hslash }^2}{2m}\frac{{\partial }^2{\psi }_1}{\partial x^2}=i\hslash \frac{\partial {\psi }_1}{\partial t},and-\frac{{\hslash }^2}{2m}\frac{{\partial }^2{\psi }_2}{\partial x^2}=i\hslash \frac{\partial {\psi }_2}{\partial t}$$
  • then, any linear combination of ${\psi }_1$ and ${\psi }_2$ is also a solution:
  $$\psi =\alpha {\psi }_1+\beta {\psi }_2\$\$\$\$-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}(\alpha {\psi }_1+\beta {\psi }_2)=i\hslash \frac{\partial }{\partial t}(\alpha {\psi }_1+\beta {\psi }_2)$$

  - $\alpha$ and $\beta$ are constants