Quantum Physics

**The Schrödinger equation:** Introduction

*Cont’d from “The Hamiltonian operator: Introduction”, see “Operators”*

In its most general sense, the Schrödinger equation is simply the eigen-equation for the Hamiltonian acting on a
wavefunction | *ψ* ⟩, given by

Now, by applying the Hamiltonian operator to the wavefunction, it becomes

which we can substitute into its eigen-equation to obtain

where ∇^{2} is the Laplacian operator. This is known as the time-independent Schrödinger equation (TISE) in 3-dimensions – often introduced as the linear TISE:

*i.e. *the TISE in 1-dimension.

Note how in both cases the potential energy is still in operator form, which can be varied depending on the scenario. For example, charged particles will have potential energy provided by the Coulomb potential

It is rare that the potential energy will take the form of a conventional operator, leading to the energy conservation form of the Hamiltonian being written simply as

though this is only a notational difference.

# Time dependency

Now, to examine the time dependence of the Schrödinger equation we must apply the Hamiltonian operator to a time dependent wavefunction, | *ψ*(**r**, *t*) ⟩. We find its eigen-equation to be

which leads to the time-dependent Schrödinger equation (TDSE)

or in 1-dimension

Solving this equation can tell us information about the time evolution of the wavefunction. This means we can find how it moves or how it interacts within the confines of quantum mechanics.

It is worth noting that the Schrödinger equation often sees a correction in the mass of the particle *m* in scenarios of interaction between two or more particles. This is best expressed by the reduced mass *μ*.