| | |
| | | \end{bmatrix} |
| | | $$ |
| | | |
| | | $$ |
| | | \begin{array}{rll} |
| | | E \psi &= H\psi & \text{Expanding the Hamiltonian Operator} \\ |
| | | &= -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} \psi + \frac{1}{2}m\omega x^2 \psi & \text{Using the ansatz $\psi(x) = e^{-kx^2}f(x)$, hoping to cancel the $x^2$ term} \\ |
| | | &= -\frac{\hbar^2}{2m} [4k^2x^2f(x)+2(-2kx)f'(x) + f''(x)]e^{-kx^2} + \frac{1}{2}m\omega x^2 f(x)e^{-kx^2} &\text{Removing the $e^{-kx^2}$ term from both sides} \\ |
| | | & \Downarrow \\ |
| | | Ef(x) &= -\frac{\hbar^2}{2m} [4k^2x^2f(x)-4kxf'(x) + f''(x)] + \frac{1}{2}m\omega x^2 f(x) & \text{Choosing $k=\frac{im}{2}\sqrt{\frac{\omega}{\hbar}}$ to cancel the $x^2$ term, via $-\frac{\hbar^2}{2m}4k^2=\frac{1}{2}m \omega$} \\ |
| | | &= -\frac{\hbar^2}{2m} [-4kxf'(x) + f''(x)] \\ |
| | | \end{array} |
| | | $$ |
| | | |
| | | > [!warn] |
| | | > Due to limitations in the [underlying parsing library](https://github.com/remarkjs/remark-math), block math in Quartz requires the `$$` delimiters to be on newlines like above. |
| | | |
| | |
| | | |
| | | .katex-display { |
| | | overflow-x: auto; |
| | | overflow-y: visible; |
| | | |
| | | & > .katex .katex-html { |
| | | .base, |
| | | .strut { |
| | | display: inline; |
| | | } |
| | | } |
| | | overflow-y: hidden; |
| | | } |