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

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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:

 

 

Rearrange:

  
  

Time-Independent Schrodinger Equation

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Del Operator:  

Derivation

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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

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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: