# Materials in Electronics/Schrödinger's Equation

Schrödinger's Equation is a differential equation that describes the evolution of *Ψ(x)* over time. By solving the differential equation for a particular situation, the wave function can be found. It is a statement of the conservation of energy of the particle.

## Schrödinger's Equation in 1-Dimension[edit | edit source]

In the simplest case, a particle in one dimension, it is derived as follows:

where

*T(x)*is the kinetic energy of the particle*V(x)*is the potential energy of the particle*E*is the energy of the particle, which is constant

Substituting for the kinetic energy of wave, as shown here:

Now we need to get this differential equation in terms of *Ψ(x)*. Assume that *Ψ(x)* is given by

Double differentiating our trial solution,:

Rearranging for *k ^{2}*

Substituting this in the differential equation gives:

Multiplying through by *Ψ(x)* gives us **Schrödinger's Equation in 1D**:

[Schrödinger's Equation in 1D]

Solving the Schrödinger Equation gives us the wavefunction of the particle, which can be used to find the electron distribution in a system.

This is a **time-independent** solution - it will not change as time goes on. It is straightforward to add time-dependence to this equation, but for the moment we will consider only time-independent wave functions, so it is not necessary. The time-dependent wavefunction is denoted by

While this equation was derived for a specific function, a complex exponential, it is more general than it appears as Fourier analysis can express any continuous function over range *L* as a sum of functions of this kind:

## The Schrödinger Equation as an Eigenequation[edit | edit source]

The Schrödinger Equation can be expressed as an **eigenequation** of the form:

[Schrödinger Equation as an Eigenequation]

where

*ψ*is the eigenfunction (or eigenstate, both mean the same thing)*E*is the eigenvalue corresponding to the energy.*H*is the Hamiltonian operator given by:

[1D Hamiltonian Operator]

This means that by applying the operator, *H*, to the function *ψ(x)*, we will obtain a solution that is simply a scalar multiple of *ψ(x)*. This multiple is *E* - the energy of the particle.

This also means that every wavefunction (i.e. every solution to the Schrödinger Equation) has a particular associated energy.

## Higher Dimensions[edit | edit source]

The equation that we just derived is the Schrödinger equation for a particle in one dimension. Adding more dimensions is not difficult. The three dimensional equation is:

Where is the Laplace operator, which, in Cartesian coordinates, is given by:

See this page for the derivation. It is also possible to add more dimensions, but this does not generally yield useful results, given that we inhabit a 3D universe.

## Spin[edit | edit source]

In order to integrate Schrödinger's equation with relativity, Paul Dirac showed that electrons have an additional property, called **spin**. This does not actually mean the electron is spinning on an axis, but in some ways it is a useful analogy.

The spin on an electron can take two values;

We can incorporate spin into the wavefunction, *Ψ* by multiplying by an addition component - the spin wavefunction, *σ(s)*, where *s* is ±1/2. This is often just called "spin-up" and "spin-down", respectively. The full, time-dependent, wavefunction is now given by:

## Conditions on the Wavefunction[edit | edit source]

In order to represent a particle's state, the wavefunction must satisfy several conditions:

- It must be square-integrable, and moreover, the integral of the wavefunction's probability density function must be equal to unity, as the electron must exist somewhere in all of space:

- For 1D systems this is:

- must be continuous, because its derivative, which in proportional to momentum, must be finite.

- must be continuous, because its derivative, which is proportional to kinetic energy, must be finite.

- must satisfy boundary conditions. In particular, as
*x*tends to infinity,*ψ(*tends to zero. (This is required to satisfy the normalistation condition above).**r**)

## Examples of Use of Schrödinger's Equation[edit | edit source]

Schrödinger's Equation can be used to find wavefunctions for many physical systems. See Confined Particles for more information.

## Summary[edit | edit source]

- Shrödinger's Equation (SE) is a statement of the Law of Conservation of Energy.
- It is given by
- By solving the equation, one can obtain the wavefunction,
*ψ*. - From the wavefunction we find the distribution of the electron's probability function.
- The probability of the electron existing over all space must be 1.
- SE gives a set of discrete wavefunctions, each with an associated energy.
- An electron cannot exist at an energy other than these.