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

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

$T\left(x\right)+V\left(x\right)=E,$

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:

${{\hbar ^{2}k^{2}} \over {2m}}+V\left(x\right)=E$

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

$\psi \left(x\right)=\psi _{0}e^{jkx}$

Double differentiating our trial solution,:

${{d^{2}} \over {dx^{2}}}\psi \left(x\right)$	$=-k^{2}\psi _{0}e^{jkx}$
	$=-k^{2}\psi \left(x\right)$

Rearranging for k²

$k^{2}=-{1 \over {\psi \left(x\right)}}{{d^{2}} \over {dx^{2}}}\psi \left(x\right)$

Substituting this in the differential equation gives:

$-{1 \over {\psi \left(x\right)}}{\hbar ^{2} \over {2m}}{{d^{2}} \over {dx^{2}}}\psi \left(x\right)+V\left(x\right)=E$

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

[Schrödinger's Equation in 1D]

$-{\hbar ^{2} \over {2m}}{{d^{2}\psi \left(x\right)} \over {dx^{2}}}+V\left(x\right)\psi \left(x\right)=E\psi \left(x\right)$

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 $\Psi \left(x,t\right)$

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:

${f}\left({x}\right)=\sum \limits _{n=1}^{\infty }{c_{n}e^{-jkx},}\quad k={{n\pi } \over L}$

The Schrödinger Equation as an Eigenequation

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

[Schrödinger Equation as an Eigenequation]

$H\psi =E\psi \,$

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]

$H=-{\hbar \over {2m}}{{d^{2}} \over {dx^{2}}}+V\left(x\right)$

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

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:

$-{{\hbar ^{2}} \over {2m}}\nabla ^{2}\psi \left({\mathbf {r} }\right)+V\left({\mathbf {r} }\right)\psi \left({\mathbf {r} }\right)=E\psi \left({\mathbf {r} }\right)$

Where $\nabla ^{2}$ is the Laplace operator, which, in Cartesian coordinates, is given by:

$\nabla ^{2}={{\partial ^{2}} \over {\partial x^{2}}}+{{\partial ^{2}} \over {\partial y^{2}}}+{{\partial ^{2}} \over {\partial z^{2}}}$

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

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;

$\pm {1 \over 2}\hbar$

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:

$\Psi \left({{\mathbf {r} },t}\right)=\psi \left({\mathbf {r} }\right)w\left(t\right)\sigma \left(s\right)$

Conditions on the Wavefunction

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:

$\int \limits _{-\infty }^{\infty }{\int \limits _{-\infty }^{\infty }{\int \limits _{-\infty }^{\infty }{\left|{\psi \left({\mathbf {r} }\right)}\right|^{2}dx\,dy\,dz}}}=1$

For 1D systems this is:

$\int \limits _{-\infty }^{\infty }{\left|{\psi \left(x\right)}\right|^{2}dx=1}$

• $\psi \left(\mathbf {r} \right)$ must be continuous, because its derivative, which in proportional to momentum, must be finite.

• ${\frac {d\psi \left(\mathbf {r} \right)}{dx}}$ must be continuous, because its derivative, which is proportional to kinetic energy, must be finite.

• $\psi \left(\mathbf {r} \right)$ must satisfy boundary conditions. In particular, as x tends to infinity, ψ(r) tends to zero. (This is required to satisfy the normalistation condition above).

Examples of Use of Schrödinger's Equation

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

Summary

• Shrödinger's Equation (SE) is a statement of the Law of Conservation of Energy.
• It is given by $-{\hbar ^{2} \over {2m}}{{d^{2}\psi \left(x\right)} \over {dx^{2}}}+V\left(x\right)\psi \left(x\right)=E\psi \left(x\right)$
• 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.