Changes

:<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 Schrödinger 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>

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

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

<math>

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

<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.

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.

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==