# Materials Science and Engineering/Equations/Quantum Mechanics

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

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- Energy:

- Frequency:

- Angular Frequency:

- Wavenumber:

- Plank's Constant:

## De Broglie Hypothesis

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- Wavelength:

- Momentum:

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

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## Time-Dependent Schrodinger Equation

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- Ket:

- Reduced Planck's Constant:

- Hamiltonian:

- The Hamiltonian describes the total energy of the system.

- Partial Derivative:

- Mass:

- Potential:

### Derivation

editBegin with a step from the time-independent derivation

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Set each side equal to a constant,

Multiply by to remove constants on the right side of the equation.

The solution is similar to what was found previously

The amplitude at a point is equal to the amplitude at a point

The following equation must be true:

Rewrite in terms of the frequency

Enter the equation into the expression of

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

To extract , differentiate with respect to time:

Rearrange:

## Time-Independent Schrodinger Equation

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- Del Operator:

### Derivation

editThe 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:

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Separate the function into two variables:

Insert the function into the wave equation:

Rearrange to separate and

Set each side equal to an arbitrary constant,

Solve this equation

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

The following condition must be true:

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Incorporate the de Broglie wave expression

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Use the symbol

Use the expression of kinetic energy,

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

## Non-Relativistic Schrodinger Wave Equation

editIn 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:

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- kinetic energy operator:
- mass of the particle:
- momentum operator:
- potential energy operator:
- real scalar function of the position operator :
- Gradient operator:
- Laplace operator: