[[Quantum Mechanics]]
==Inner Product==: $$\psi^{\hat{\dagger}}\psi$$
This is the inner product of a vector, in this case we have written then wavefunction as a vector. This can also be written like
$$\Braket{\psi|\psi}$$
Where $\bra{\psi}$ is the complex conjugate of vector $\ket{\psi}$
==Hamiltonian==: $$H =\frac{p^{2}}{2m} + V(x)$$This is the Hamiltonian, a KE term and a potential term — with potential dependent on a specific function ($V(x)$) which can be given in a particular setup/problem.
==Schrodinger’s Equation:==
$$ih\frac{\partial}{\partial t}\psi(r,t) = -\frac{h^{2}}{2m}\nabla^
{2}\psi(r,t) + V(r,t)\psi(x,t)$$
This is the time dependent Schrodinger equation whereas the left side of the equation is the partial WRT to time and the right hand side consists of the spatial derivative. This will be the basis for all #TISE breakdowns.
==Momentum Operator:==
The #momentum #operator for standard wave functions is as written$$\hat{p} = -i \hbar \frac{d}{dx}$$
This will be expanded in relativistic quantum mechanics, and QED with #vector-potentials etc.
Note that $\hat{p}^{2}$ is $\left (-i \hbar \frac{d}{dx}\psi(x) \right )^{2}$ which is $$-\frac{\hbar^{2}}{2m} \frac{d^{2}}{dx^{2}} = -\frac{\hbar^{2}}{2m}\nabla^{2}$$
==Statistical Approach:==
Wan to find the probability of a wave function being located within a certain position$$\int_{a}^{b} |\psi(x,t)|^{2}dx = \int_{a}^{b}\psi^{\dagger}\psi dx$$ which is the probability of finding the wavefunction between positions $a$ and $b$ at time $t$
