Quantum Physics

The Schrödinger Equation in its Various Forms

Perhaps no equation in quantum mechanics is as ubiquitous as the Schrödinger equation.

However, the actual form the equation takes can vary greatly depending on the physical system or situation it aims to describe. Often, this can be a source of confusion for those starting out in quantum mechanics. Therefore, in this article we will derive, explain, and relate these various representations.

Table Of Contents

1. Discrete Quantum States
   a. Quantum Matrix Mechanics
2. Continuous Quantum States
   a. Quantum Wave Mechanics
3. The Hamiltonian
4. The Time-Dependent Schrödinger Equation
5. Deriving the Time-Independent Schrödinger Equation
   a. Separation of Variables
   b. The Time-Dependent Differential Equation
   c. The Time-Independent Schrödinger Equation
   d. Solutions in a Continuous Basis
   e. Solutions in a Discrete Basis
   f. More General Solutions
6. Conclusion

The Schrödinger equation is a differential equation which describes how the quantum state of a system evolves in time. This quantum state 𝚿 could be in a discrete basis or a continuous basis. Before we dive into the Schrödinger equation we first need to clarify the difference between these two representations.

Discrete Quantum States