Materials Science and Engineering/Equations/Quantum Mechanics

From Wikiversity

Jump to: navigation, search

Contents

[edit] Relation between energy and frequency of a quanta of radiation

 E=hf\;
 E=\hbar \omega

 \mathbf{p}=\hbar \mathbf{k}\;
Energy: E
Frequency: f
Angular Frequency: ω = 2πf
Wavenumber: k = 2π / λ
Plank's Constant: h

[edit] De Broglie Hypothesis

 p=h/\lambda\;
Wavelength: λ
Momentum: p

[edit] Phase of a Plane Wave Expressed as a Complex Phase Factor

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

[edit] Time-Dependent Schrodinger Equation

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

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

[edit] Derivation

Begin with a step from the time-independent derivation

 \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, − κ2

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

Multiply by c2 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 + τ

N e^{\pm i \beta t} = N e^{\pm i \beta (t + \tau)}

The following equation must be true:

\beta \tau = 2 \pi \;

Rewrite β in terms of the frequency

\beta = 2 \pi v \;

Enter the equation into the expression of ζ

\zeta (t) = Ne^{\pm 2 \pi i v t}

\zeta (t) = N e^{-i E t / \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:

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

[edit] Time-Independent Schrodinger Equation

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

  H \left|\psi_n\right\rang = E_n \left|\psi_n \right\rang.
Del Operator: \nabla

[edit] 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:

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

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 Ψ and ζ

\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, − κ2

\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:

N e^{\pm i \kappa x} = N e^{\pm i \kappa (x + \lambda)}

The following condition must be true:

 \kappa \lambda = 2 \pi\;

Incorporate the de Broglie wave expression

 \frac{h}{mv} = \lambda

\kappa = \frac{2 \pi m v}{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

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

[edit] 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:

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


 
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)


 
H \psi\left(\mathbf{r}, t\right) = \mathrm{i} \hbar \frac{\partial \psi}{\partial t} \left(\mathbf{r}, t\right)
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
Retrieved from "http://en.wikiversity.org/wiki/Materials_Science_and_Engineering/Equations/Quantum_Mechanics"