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The '''Schrodinger 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 Schrodinger]].
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==
==Mathematical forms==
===General time-dependent form===
===General time-dependent form===
The Schrodinger equation may generally be written
The Schrödinger equation may generally be written
<math>i\hbar\frac{\partial}{\partial t}|\Psi\rangle=\hat H|\Psi\rangle</math>
<math>i\hbar\frac{\partial}{\partial t}|\Psi\rangle=\hat H|\Psi\rangle</math>
:<math>\hat H</math> is the [[Hamiltonian]] operator.
:<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 Schrodinger equation can be written
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>
<math>
</math>
</math>
This can be expanded to give the three dimensional time dependent Schrodinger equation using the [[Laplacian|Laplacian operator]], <math>\nabla^2</math>:
This can be expanded to give the three dimensional time dependent Schrödinger equation using the [[Laplacian|Laplacian operator]], <math>\nabla^2</math>:
<math>
<math>
===Derivation===
===Derivation===
The quickest and easiest way to derive Schrodinger's equation is to understand the Hamiltonian ''operator'' in quantum mechanics. In classical mechanics, the total energy of a system is given by
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>
<math>
\hat H = \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x) </math>
\hat H = \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x) </math>
from which Schrodinger's equation and the eigenvalue problem <math> \hat H\Psi = E\Psi</math> can be easily seen.
from which Schrödinger's equation and the eigenvalue problem <math> \hat H\Psi = E\Psi</math> can be easily seen.
===Time-independent Schrodinger Equation===
===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 Schrodinger 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>
<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 Schrodinger equation''':
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>- \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.
<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 Schrodinger Equation===
===Solutions to the Schrödinger Equation===
Solutions to the Schrodinger 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]].
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 Schrodinger 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.
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===
===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 Schrodinger equation is time-independent and is often written
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>
<math>E\psi=\hat H\psi</math>
===Interpretation of the Wavefunction===
===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]].
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====
====Normalisation====
Wavefunctions are said to be "normalisable" if the following integral exists:
Wavefunctions are said to be "normalisable" if the following integral exists:
==Examples for the Time-Independent Equation==
==Examples for the Time-Independent Equation==
===Free particle in one dimension===
===Free particle in one dimension===
In this case, <math>V(x)=0</math>, the time-independent Schrodinger equation is:
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>
<math>\frac{d^2 \psi}{dx^2} = - \frac{2mE}{\hbar^2} \psi</math>
===Particle in a box===
===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:
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:
This is also known as "the infinite square well".
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 Schrodinger equation) that
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>
<math>\psi=\sin(kx)</math>
===Derivation===
===Derivation===
The particle cannot exits 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 Schrodinger equation inside the potential well becomes:
The particle cannot exits 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>
<math>- \frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} = E \psi</math>
<math>E_n=\frac{\hbar^2n^2\pi^2}{2m}</math>
<math>E_n=\frac{\hbar^2n^2\pi^2}{2m}</math>
You may wonder what happens about the constant <math>B</math>. Since the Schrodinger 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:
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) =
<math>V(x) =
==Relativity==
==Relativity==
The Schrodinger equation is non-relativistic in nature. It can be re-derived taking [[special relativity]] into account. This results in the Klein-Gordon equation.
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==
==See also==