# Materials Science and Engineering/Equations/Quantum Mechanics

## Relation between energy and frequency of a quanta of radiation

 ${\displaystyle E=hf\;}$
${\displaystyle E=\hbar \omega }$

 ${\displaystyle \mathbf {p} =\hbar \mathbf {k} \;}$

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

## De Broglie Hypothesis

 ${\displaystyle p=h/\lambda \;}$

Wavelength: ${\displaystyle \lambda }$
Momentum: ${\displaystyle p}$

## Phase of a Plane Wave Expressed as a Complex Phase Factor

 ${\displaystyle \psi \approx e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}}$


## Time-Dependent Schrodinger Equation

 ${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi +V\Psi \;}$

 ${\displaystyle \mathrm {i} \hbar {\frac {d}{dt}}\left|\psi \left(t\right)\right\rangle =H(t)\left|\psi \left(t\right)\right\rangle }$

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

### Derivation

Begin with a step from the time-independent derivation

 ${\displaystyle {\frac {1}{\Psi }}{\frac {d^{2}\Psi }{dx^{2}}}={\frac {1}{c^{2}\zeta }}{\frac {d^{2}\zeta }{dt^{2}}}}$


Set each side equal to a constant, ${\displaystyle -\kappa ^{2}}$

${\displaystyle -\kappa ^{2}={\frac {1}{c^{2}\zeta }}{\frac {d^{2}\zeta }{dt^{2}}}}$

Multiply by ${\displaystyle c^{2}}$ to remove constants on the right side of the equation.

${\displaystyle -\beta ^{2}={\frac {1}{\zeta }}{\frac {d^{2}\zeta }{dt^{2}}}}$

The solution is similar to what was found previously

${\displaystyle \zeta (t)=Ne^{\pm i\beta t}}$

The amplitude at a point ${\displaystyle t}$ is equal to the amplitude at a point ${\displaystyle t+\tau }$

${\displaystyle Ne^{\pm i\beta t}=Ne^{\pm i\beta (t+\tau )}}$

The following equation must be true:

${\displaystyle \beta \tau =2\pi \;}$

Rewrite ${\displaystyle \beta }$ in terms of the frequency

${\displaystyle \beta =2\pi v\;}$

Enter the equation into the expression of ${\displaystyle \zeta }$

${\displaystyle \zeta (t)=Ne^{\pm 2\pi ivt}}$

${\displaystyle \zeta (t)=Ne^{-iEt/\hbar }}$

The time-dependent Schrodinger equation is a product of two 'sub-functions'

${\displaystyle \Psi (x,t)=\psi (x)\zeta (t)\;}$

${\displaystyle \Psi (x,t)=\psi e^{-iEt/\hbar }}$

To extract ${\displaystyle E}$, differentiate with respect to time:

${\displaystyle {\frac {\partial \Psi }{\partial t}}={\frac {-iE}{\hbar }}\psi e^{-iEt/\hbar }}$

${\displaystyle {\frac {\partial \Psi }{\partial t}}={\frac {E}{i\hbar }}\psi e^{-iEt/\hbar }}$

Rearrange:

 ${\displaystyle i\hbar {\frac {\partial \Psi }{\partial t}}=E\Psi }$
${\displaystyle {\hat {H}}\Psi =E\Psi }$


## Time-Independent Schrodinger Equation

 ${\displaystyle H\Psi =E\Psi \;}$

${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi (x)}{dx^{2}}}+U(x)\psi (x)=E\psi (x)}$
${\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+U(\mathbf {r} )\right]\psi (\mathbf {r} )=E\psi (\mathbf {r} )}$
 ${\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +(U-E)\psi =0}$

 ${\displaystyle H\left|\psi _{n}\right\rangle =E_{n}\left|\psi _{n}\right\rangle .}$

Del Operator: ${\displaystyle \nabla }$

### 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:

 ${\displaystyle {\frac {d^{2}z}{dx^{2}}}={\frac {1}{c^{2}}}{\frac {d^{2}z}{dt^{2}}}}$


Separate the function into two variables:

${\displaystyle z(x,t)=\Psi (x)\zeta (t)\;}$

Insert the function into the wave equation:

${\displaystyle \zeta {\frac {d^{2}\Psi }{dx^{2}}}={\frac {\Psi }{c^{2}}}{\frac {d^{2}\zeta }{dt^{2}}}}$

Rearrange to separate ${\displaystyle \Psi }$ and ${\displaystyle \zeta }$

${\displaystyle {\frac {1}{\Psi }}{\frac {d^{2}\Psi }{dx^{2}}}={\frac {1}{c^{2}\zeta }}{\frac {d^{2}\zeta }{dt^{2}}}}$

Set each side equal to an arbitrary constant, ${\displaystyle -\kappa ^{2}}$

${\displaystyle {\frac {1}{\Psi }}{\frac {d^{2}\Psi }{dx^{2}}}=-\kappa ^{2}}$

${\displaystyle {\frac {d^{2}\Psi }{dx^{2}}}=-\kappa ^{2}\Psi }$

Solve this equation

${\displaystyle \Psi (x)=Ne^{\pm i\kappa x}}$

The amplitude at one point needs to be equal to the amplitude at another point:

${\displaystyle Ne^{\pm i\kappa x}=Ne^{\pm i\kappa (x+\lambda )}}$

The following condition must be true:

 ${\displaystyle \kappa \lambda =2\pi \;}$


Incorporate the de Broglie wave expression

 ${\displaystyle {\frac {h}{mv}}=\lambda }$


${\displaystyle \kappa ={\frac {2\pi mv}{h}}}$

Use the symbol ${\displaystyle \hbar }$

${\displaystyle \hbar ={\frac {h}{2\pi }}}$

${\displaystyle {\frac {d^{2}\Psi }{dx^{2}}}=-{\frac {m^{2}v^{2}}{\hbar ^{2}}}\Psi }$

${\displaystyle {\frac {-\hbar ^{2}}{m^{2}v^{2}}}{\frac {d^{2}\Psi }{dx^{2}}}=\Psi }$

Use the expression of kinetic energy, ${\displaystyle E_{kinetic}={\frac {1}{2}}mv^{2}}$

${\displaystyle {\frac {-\hbar ^{2}}{2m}}{\frac {d^{2}\Psi }{dx^{2}}}=E\Psi }$

Modify the equation by adding a potential energy term and the Laplacian operator

 ${\displaystyle {\frac {-\hbar ^{2}}{2m}}\nabla ^{2}\Psi +V\Psi =E\Psi }$
${\displaystyle {\hat {H}}\Psi =E\Psi }$


## 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:

 ${\displaystyle H\psi \left(\mathbf {r} ,t\right)=\left(T+V\right)\psi \left(\mathbf {r} ,t\right)}$

 ${\displaystyle H\psi \left(\mathbf {r} ,t\right)=\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\left(\mathbf {r} \right)\right]\psi \left(\mathbf {r} ,t\right)}$

 ${\displaystyle H\psi \left(\mathbf {r} ,t\right)=\mathrm {i} \hbar {\frac {\partial \psi }{\partial t}}\left(\mathbf {r} ,t\right)}$

kinetic energy operator: ${\displaystyle T={\frac {p^{2}}{2m}}}$
mass of the particle: ${\displaystyle m\;}$
momentum operator: ${\displaystyle \mathbf {p} =-\mathrm {i} \hbar \nabla }$
potential energy operator: ${\displaystyle V=V\left(\mathbf {r} \right)}$
real scalar function of the position operator ${\displaystyle \mathbf {r} }$: ${\displaystyle V}$
Gradient operator: ${\displaystyle \nabla }$
Laplace operator: ${\displaystyle \nabla ^{2}}$