Materials Science and Engineering/Equations/Quantum Mechanics

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

Angular Frequency:
Plank's Constant:

De Broglie Hypothesis[edit | edit source]


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


Time-Dependent Schrodinger Equation[edit | edit source]

Reduced Planck's Constant:
The Hamiltonian describes the total energy of the system.
Partial Derivative:

Derivation[edit | edit source]

Begin with a step from the time-independent derivation


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:



Time-Independent Schrodinger Equation[edit | edit source]

Del Operator:

Derivation[edit | edit source]

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:

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:


Incorporate the de Broglie wave expression


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[edit | edit source]

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