Changes
Jump to navigation
Jump to search
Line 8:
Line 8: − + + + + + Line 16:
Line 20: − Note that solutions may be [[complex numbers|complex]] in nature.+ + + + + + + + + Line 35:
Line 47: + + + + + + + + + + + + + + + + + + Line 46:
Line 76: − + Line 64:
Line 94: − + − + + + + + − + − with energy given by+ − + − + Line 125:
Line 159: − +
Schrodinger equation (view source)
Revision as of 18:32, September 20, 2016
, 18:32, September 20, 2016Added section on time-independent equation and on one of the examples
<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>
where <math>i</math> is the [[complex number|imaginary unit]],<br><math>\hbar</math> is [[Planck's constant]] divided by <math>2\pi</math>, <br><math>|\Psi\rangle</math> is the quantum mechanical state or wavefunction (expressed here in [[Dirac notation]]), and <br><math>\hat H</math> is the [[Hamiltonian]] operator.
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 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 Schrodinger equation can be written
</math>
</math>
This can be expanded to give the three dimensional time dependent Schrodinger 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===
===Derivation===
from which Schrodinger's equation and the eigenvalue problem <math> \hat H\Psi = E\Psi</math> can be easily seen.
from which Schrodinger's equation and the eigenvalue problem <math> \hat H\Psi = E\Psi</math> can be easily seen.
===Time-independent 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 know 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:
<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''':
<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 Schrodinger 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]].
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.
===Eigenvalue problems===
===Eigenvalue problems===
===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.
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====
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.
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==
==Examples for the Time-Independent Equation==
===Free particle in one dimension===
===Free particle in one dimension===
In this case, <math>V(x)=0</math> and so we see that the solution to the Schrodinger equation must be
In this case, <math>V(x)=0</math>, the time-independent Schrodinger 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=Ae^{-ikx}</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>
<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.
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===
===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:
<math>U(x) =
<math>V(x) =
\begin{cases}
\begin{cases}
\infty &\quad\text{if } x < 0\\
\infty &\quad\text{if } x < 0\\
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 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:
<math>U(x) =
<math>V(x) =
\begin{cases}
\begin{cases}
\sqrt{\frac{2}{a}} \sin{\omega x} &\quad\text{if } x < 0\\
\sqrt{\frac{2}{a}} \sin{\omega x} &\quad\text{if } x < 0\\
(username removed)