Changes

===Time-independent Schrodinger Equation===

===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:

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:

<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>

===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====

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

<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 ind the value of the the constants, we substitute in boundary conditions. An additional constraint is that wavefunctions must be continious (they cannot suddenly change value). Hence

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>

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

<math>V(x) =  

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 Schrodinger 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==

*[[Quantum Mechanics]]

*[[Quantum Mechanics]]

*[[Differential equation]]

*[[Differential equation]]