Materials Science and Engineering/Equations/Quantum Mechanics

Relation between energy and frequency of a quanta of radiation edit

Angular Frequency:  
Plank's Constant:  

De Broglie Hypothesis edit


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


Time-Dependent Schrodinger Equation edit

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

Derivation edit

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

Del Operator:  

Derivation edit

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

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: