PX156 - C2 - matter waves

planck:

E = h f = ℏ ω

de broglie:

p = \frac{h}{λ} = ℏ k

E = \frac{ℏ^{2} k^{2}}{2 m} = \frac{p^{2}}{2 m} = K E

$ψ = \cos (k x - ω t)$ is not a solution of TDSE

normalization condition

\begin{aligned} ψ & = ψ_{1} + i ψ_{2} \\ | ψ | & = \sqrt{ψ_{2}^{2} + ψ_{2}^{2}} \\ | ψ |^{2} & = ψ_{1}^{2} + ψ_{2}^{2} \\ = ψ ψ^{*} \\ = (ψ_{1} + i ψ_{2}) (ψ_{1} - i ψ_{2}) \end{aligned}

max born proposed:

\int_{a}^{b} | ψ (x, t) |^{2} d x = \int_{a}^{b} ψ (x, t) ψ^{*} (x, t) d x

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

the normalization condition:

\int_{a l l s p a c e} | ψ (x, t) |^{2} d x = 1

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

plane wave:

| ψ |^{2} = ψ ψ^{*} = A \exp (i (k x - ω t)) \cdot A \exp (- i (k x - ω t)) = A^{2}

therefore plane waves cannot represent matter waves
solution: wave packet
- $ψ_{1}$ and $ψ_{2}$ are the solutions of the TDSE:
$- \frac{ℏ^{2}}{2 m} \frac{\partial^{2} ψ_{1}}{\partial x^{2}} = i ℏ \frac{\partial ψ_{1}}{\partial t}, a n d - \frac{ℏ^{2}}{2 m} \frac{\partial^{2} ψ_{2}}{\partial x^{2}} = i ℏ \frac{\partial ψ_{2}}{\partial t}$
- then, any linear combination of $ψ_{1}$ and $ψ_{2}$ is also a solution:
$ψ = α ψ_{1} + β ψ_{2} $ $ $ $ - \frac{ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} (α ψ_{1} + β ψ_{2}) = i ℏ \frac{\partial}{\partial t} (α ψ_{1} + β ψ_{2})$
- $α$ and $β$ are constants