# Materials Science and Engineering/Equations/Quantum Mechanics
## Relation between energy and frequency of a quanta of radiation

    Energy: $E$
    Frequency: $f$
    Angular Frequency: $\omega =2\pi f$
    Wavenumber: $k=2\pi /\lambda$
    Plank's Constant: $h$

## De Broglie Hypothesis

    Wavelength: $\lambda$
    Momentum: $p$

## Time-Dependent Schrodinger Equation

    Ket: $|\psi (t)\rangle$
    Reduced Planck's Constant: $\hbar$
    Hamiltonian: $H(t)$
       The Hamiltonian describes the total energy of the system.
    Partial Derivative: $\partial /\partial t$
    Mass: $m$
    Potential: $V$

### Derivation

Begin with a step from the time-independent derivation
Set each side equal to a constant, $-\kappa ^{2}$
$-\kappa ^{2}={\frac {1}{c^{2}\zeta }}{\frac {d^{2}\zeta }{dt^{2}}}$
Multiply by $c^{2}$ to remove constants on the right side of the equation.
$-\beta ^{2}={\frac {1}{\zeta }}{\frac {d^{2}\zeta }{dt^{2}}}$
The solution is similar to what was found previously
$\zeta (t)=Ne^{\pm i\beta t}$
The amplitude at a point $t$ is equal to the amplitude at a point $t+\tau$
$Ne^{\pm i\beta t}=Ne^{\pm i\beta (t+\tau )}$
The following equation must be true:
$\beta \tau =2\pi \;$
Rewrite $\beta$ in terms of the frequency
$\beta =2\pi v\;$
Enter the equation into the expression of $\zeta$
$\zeta (t)=Ne^{\pm 2\pi ivt}$
$\zeta (t)=Ne^{-iEt/\hbar }$
The time-dependent Schrodinger equation is a product of two 'sub-functions'
$\Psi (x,t)=\psi (x)\zeta (t)\;$
$\Psi (x,t)=\psi e^{-iEt/\hbar }$
To extract $E$, differentiate with respect to time:
${\frac {\partial \Psi }{\partial t}}={\frac {-iE}{\hbar }}\psi e^{-iEt/\hbar }$
${\frac {\partial \Psi }{\partial t}}={\frac {E}{i\hbar }}\psi e^{-iEt/\hbar }$
Rearrange:

## Time-Independent Schrodinger Equation

    $-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi (x)}{dx^{2}}}+U(x)\psi (x)=E\psi (x)$
    $\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+U(\mathbf {r} )\right]\psi (\mathbf {r} )=E\psi (\mathbf {r} )$
    Del Operator: $\nabla$

### Derivation

The Schrodinger Equation is based on two formulas:
* The classical wave function derived from the Newton's Second Law
* The de Broglie wave expression
Formula of a classical wave:
Separate the function into two variables:
$z(x,t)=\Psi (x)\zeta (t)\;$
Insert the function into the wave equation:
$\zeta {\frac {d^{2}\Psi }{dx^{2}}}={\frac {\Psi }{c^{2}}}{\frac {d^{2}\zeta }{dt^{2}}}$
Rearrange to separate $\Psi$ and $\zeta$
${\frac {1}{\Psi }}{\frac {d^{2}\Psi }{dx^{2}}}={\frac {1}{c^{2}\zeta }}{\frac {d^{2}\zeta }{dt^{2}}}$
Set each side equal to an arbitrary constant, $-\kappa ^{2}$
${\frac {1}{\Psi }}{\frac {d^{2}\Psi }{dx^{2}}}=-\kappa ^{2}$
${\frac {d^{2}\Psi }{dx^{2}}}=-\kappa ^{2}\Psi$
Solve this equation
$\Psi (x)=Ne^{\pm i\kappa x}$
The amplitude at one point needs to be equal to the amplitude at another point:
$Ne^{\pm i\kappa x}=Ne^{\pm i\kappa (x+\lambda )}$
The following condition must be true:
Incorporate the de Broglie wave expression
$\kappa ={\frac {2\pi mv}{h}}$
Use the symbol $\hbar$
$\hbar ={\frac {h}{2\pi }}$
${\frac {d^{2}\Psi }{dx^{2}}}=-{\frac {m^{2}v^{2}}{\hbar ^{2}}}\Psi$
${\frac {-\hbar ^{2}}{m^{2}v^{2}}}{\frac {d^{2}\Psi }{dx^{2}}}=\Psi$
Use the expression of kinetic energy, $E_{kinetic}={\frac {1}{2}}mv^{2}$
${\frac {-\hbar ^{2}}{2m}}{\frac {d^{2}\Psi }{dx^{2}}}=E\Psi$
Modify the equation by adding a potential energy term and the Laplacian operator

## Non-Relativistic Schrodinger Wave Equation

In non-relativistic quantum mechanics, the Hamiltonian of a particle can be expressed as the sum of two operators, one corresponding to kinetic energy and the other to potential energy. The Hamiltonian of a particle with no electric charge and no spin in this case is:
    kinetic energy operator: $T={\frac {p^{2}}{2m}}$
    mass of the particle: $m\;$
    momentum operator: $\mathbf {p} =-\mathrm {i} \hbar \nabla$
    potential energy operator: $V=V\left(\mathbf {r} \right)$
    real scalar function of the position operator $\mathbf {r}$: $V$
    Gradient operator: $\nabla$
    Laplace operator: $\nabla ^{2}$