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The '''Schrödinger equation''' is a linear [[differential equation]] used in various fields of [[physics]] to describe the time evolution of quantum states.  It is the fundamental equation of [[quantum mechanics|non-relativistic quantum meachincs]].  The equation is named after its discoverer, [[Erwin Schrödinger]].

==Mathematical forms==
===General time-dependent form===

The Schrödinger equation may generally be written

<math>i\hbar\frac{\partial}{\partial t}|\Psi\rangle=\hat H|\Psi\rangle</math>

where
:<math>i</math> is the [[complex number|imaginary unit]]
:<math>\hbar</math> is the [[Planck's constant|reduced Planck's constant]]
:<math>|\Psi\rangle</math> is the quantum mechanical state or wavefunction (expressed here in [[Dirac notation]])
:<math>\hat H</math> is the [[Hamiltonian]] operator.  

The left side of the equation describes how the wavefunction changes with time; the right side is related to its energy. For the simplest case of a particle of mass ''m'' moving in a one-dimensional potential ''V''(''x''), the Schrödinger equation can be written

<math>
-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V(x)\psi=i\hbar\frac{\partial \psi}{\partial t}
</math>

This can be expanded to give the three dimensional time dependent Schrödinger equation using the [[Laplacian|Laplacian operator]], <math>\nabla^2</math>:

<math>
-\frac{\hbar^2}{2m}
\nabla^2 \Psi
+V(x)\Psi=i\hbar\frac{\partial \Psi}{\partial t}
</math>

In this case, the wavefunction is <math>\Psi(x,y,z,t)</math>.


===Derivation===

The quickest and easiest way to derive Schrödinger's equation is to understand the Hamiltonian ''operator'' in quantum mechanics. In classical mechanics, the total energy of a system is given by

<math>
E = \frac{p^2}{2m} + V(x)</math>

where ''p'' is the momentum of the particle and ''V''(''x'') is its potential energy. Applying the quantum mechanical ''[[operator]]'' for momentum:

<math>
p = \frac{\hbar}{i}\frac{\partial}{\partial x} </math>

and subbing into the classical mechanical form for energy, we get the same Hamiltonian operator in quantum mechanics:

<math>
\hat H = \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x) </math> 

from which Schrödinger's equation and the eigenvalue problem <math> \hat H\Psi = E\Psi</math> can be easily seen.


===Time-independent Schrödinger Equation===

The wavefunction is a multi-variable function, that is to say it depends on both [[time]] and space. Suppose that we have a function only in terms of space, <math>\psi(x)</math>. We can write the full wavefunction as <math>\Psi(x,t) = \psi(x) e^{-iEt/ \hbar}</math> for a state of definite energy. This type of state is also known as a "stationary state" as the probability distribution for this type of wavefunction over time is constant (see below). Substituting this wavefunction into the Schrödinger equation:

<math>- \frac{\hbar^2}{2m} \frac{\partial^2 \psi(x)}{\partial x^2} e^{-iEt/ \hbar} + V(x) \psi(x) e^{-iEt/ \hbar} = E \psi(x) e^{-iEt/ \hbar} </math>

Cancelling the factor of <math>e^{-iEt/ \hbar}</math> results in a [[differential equation]] that only depends on <math>x</math>. This is the '''time-independent Schrödinger equation''':

<math>- \frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x) \psi(x) = E \psi(x)</math>

<math>\psi(x)</math> is often called the "wavefunction", though strictly speaking it is <math>\Psi(x,t)</math> that is the wavefunction.


===Solutions to the Schrödinger Equation===

Solutions to the Schrödinger equation are known as "wavefunctions". However, for a solution to be physically valid, two conditions must be imposed on these solutions. These are that both the wavefunction and its first derivative must be [[Continuous function|continuous]]. Wavefunctions may be [[Real number|real]] or [[Complex numbers|complex]].

Since the Schrödinger equation is [[Linear equations|linear]], if <math>\psi_1 (x)</math> and <math>\psi_2 (x)</math> are solutions, then <math>\psi_1 (x) + \psi_2 (x)</math> is also a solution.


===Eigenvalue problems===

In many instances, steady-state solutions to the equation are of great interest. Physically, these solutions correspond to situations in which the wavefunction has a well-defined [[energy]].  The energy is then said to be an [[eigenvalue]] for the equation, and the wavefunction corresponding to that energy is called an [[eigenfunction]] or [[eigenstate]].  In such cases, the Schrödinger equation is time-independent and is often written

<math>E\psi=\hat H\psi</math>

Here, ''E'' is energy, ''H'' is once again the Hamiltonian operator, and <math>\psi</math> is the energy eigenstate for ''E''.

One example of this type of eigenvalue problem is an electrons bound inside an [[atom]].


===Interpretation of the Wavefunction===

The wavefunction is related to the probability of observing a particle at that location in space. More specifically, where the wavefunction has a greater value of <math>|\Psi|^2</math> at one location, ''a'', compared to another, ''b'', then the particle is more likely to be observed at ''a'' than ''b''. Note that the [[Absolute value|modulus]] must be taken since the wavefunction may be [[Complex numbers|complex]].

====Normalisation====

Wavefunctions are said to be "normalisable" if the following integral exists:

<math>\int^{\infty}_{- \infty} |\Psi(x, t)|^2 dx </math>

If that integral exists, then the wave function may be normalised by introducing a constant, <math>c</math>, such that 

<math>\int^{\infty}_{- \infty} |c \Psi(x, t)|^2 dx = |c|^2 \int^{\infty}_{- \infty} |\Psi(x, t)|^2 dx = 1  </math>

Note that <math>c</math> may be complex. This will produce and new wavefunction, <math>c \Psi(x,t)</math>, which has the property that:

<math>P(a<x<b) = \int^{a}_{b} |c \Psi(x, t)|^2 dx</math>

This is the '''probability''' of observing the particle between points ''a'' and ''b''. For some wavefunctions, such as that of a free particle, the integral above does not exist and so they are said to be non-normalisable.


==Examples for the Time-Independent Equation==

===Free particle in one dimension===

In this case, <math>V(x)=0</math>, the time-independent Schrödinger equation is:

<math>\frac{d^2 \psi}{dx^2} = - \frac{2mE}{\hbar^2} \psi</math>

In this case, the solution is sinusoidal for some constants, <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>:

<math>\psi=A \sin{kx} + B \cos{kx} = Ce^{ikx} + De^{-ikx}</math>

The [[energy]] can then be found as:

<math>E=\frac{\hbar^2 k^2}{2m}</math>

Physically, this corresponds to a wave travelling with a [[momentum]] given by <math>\hbar k</math>, where k can in principle take any value. This wavefunction is not normalisable.


===Particle in a box===

Consider a one-dimensional box of width a, where the potential energy is 0 inside the box and infinite outside of it. We can describe this potential by:

<math>V(x) = 
\begin{cases}
       \infty &\quad\text{if } x < 0\\
       0 &\quad\text{if } 0 \le x \le a \\
       \infty &\quad\text{if } x > a \ 
\end{cases}
</math>

This is also known as "the infinite square well".
This means that <math>\psi</math> must be zero outside the box. One can verify (by substituting into the Schrödinger equation) that 

<math>\psi=\sin(kx)</math>

is a solution if <math>k=n\pi</math> where n is any integer. Thus, rather than the continuum of solutions for the free particle, for the particle in a box there is a set of discrete solutions with energies given by

<math>E_n=\frac{\hbar^2 k^2}{2m}=\frac{\hbar^2n^2\pi^2}{2m}</math>


===Derivation===

The particle cannot exist outside the box where the potential is infinite. Hence the wavefunction, <math>\psi</math>, here must be zero outside the box. Now let us consider inside the box. The Schrödinger equation inside the potential well becomes:

<math>- \frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} = E \psi</math>

Rearranging this equation give
<math>\frac{d^2 \psi}{dx^2} = - \frac{2mE}{\hbar^2} \psi</math>

This has the same form as the basic equation for simple harmonic motion. Hence we can give the solution as

<math>\psi (x) = A \cos{\omega x} + B \sin{\omega x}</math>

where <math>A</math> and <math>B</math> are unknown constants and <math>\omega</math> is equal to <math>\sqrt{\frac{2mE}{\hbar^2}}</math>. To indicate the value of the constants, we substitute in boundary conditions. An additional constraint is that wavefunctions must be continious (they cannot suddenly change value). Hence

<math>\psi (0) = 0 = A \cos{\omega 0} + B \sin{\omega 0} </math>

From this we can see that <math>A</math> must be zero. Using the continuous constraint again, we see that 

<math>\psi (a) = 0 = B \sin{\omega a} </math>

The continuous constraint is only satisfied when <math>\omega a = n \pi</math> where <math>n</math> is an [[integer]]. We also note that <math>n=0</math> is not a solution as the wavefunction would be zero everywhere and so the probability of finding the particle is 0. We also discard solutions for <math>n < 0 </math> since these are not "new" solutions. As <math> \sin{-n x} =-\sin{x}</math>, these are just the negative of solutions with positive <math>n</math>. By setting <math>\omega a = n \pi</math>, we can find the energy <math>E_n</math> as:

<math>\sqrt{\frac{2m E_n}{\hbar^2}} = n \pi</math>

And so:
<math>E_n=\frac{\hbar^2n^2\pi^2}{2m}</math>

You may wonder what happens about the constant <math>B</math>. Since the Schrödinger equation is linear, <math>B</math> can take any value and the resulting wavefunction would be a solution. Since <math>|\psi (x)|^2</math> represents the probability of finding the particle at position <math>x</math> and across all space, this probability must equal 1, we can solve for <math>B</math>. We do this by integrating <math>|\psi (x)|^2</math> from <math>-\infty</math> to <math> + \infty</math>. Therefore, the full solution is:

<math>V(x) = 
\begin{cases}
       \sqrt{\frac{2}{a}} \sin{\omega x} &\quad\text{if } x < 0\\
       0 &\quad\text{otherwise} \\
\end{cases}
</math>

where <math>\omega</math> is as above.


==Relativity==

The Schrödinger equation is non-relativistic in nature. It can be re-derived taking [[special relativity]] into account. This results in the Klein-Gordon equation.

==See also==
*[[Quantum Mechanics]]
*[[Differential equation]]

[[Category:Quantum Mechanics]]