Wavefunctions and the Born Interpretation
The wave-particle duality of matter is dealt with in quantum mechanics by considering that, rather than a particle traveling along a definite path, it is distributed through space like a wave. The classical idea of a trajectory is thus replaced in quantum mechanics by a wave, which is defined by a wavefunctionrepresented by ψ.
i.e. the spatial distribution of a particle is represented by a wave, and the wavefunction is the mathematical function that describes this wave. The wavefunction contains all the information that would be available from the trajectory.
From this, it follows that it must be possible to write a function that describes the complete state of a system (as it is possible to write a wavefunction for all the individual particles that make up the system). A function that completely describes a system is also called a wavefunction. (By “completely describes” we mean that the function contains information about all the properties of the system that may be experimentally determined, for example at the simplest level the position and momentum of particles.)
It is a fundamental principle in quantum mechanics that the wavefunction of a system contains all the information about measurable physical properties of the system.
The wavefunction is the function ψ described as “a function that precisely describes the system of interest ” in the section on principles of quantum mechanics on the previous page.
In general, a wavefunction is a function of the spatial coordinates of the particles that make up the system and of the time. If the way the system changes with time is not of interest, then this part of the wavefunction may usually be neglected. (It commonly proves possible to write the wavefunction as a product of a solely time-dependent part and a time-independent part.)
The wavefunction may also depend upon a property of the particles themselves known as their spin, but discussion of this will be left till later.
Although, as we have said, it is theoretically possible to write a wavefunction for any system, the function rapidly becomes exceedingly complicated. It will thus normally be the case that a fully written out wavefunction will only be encountered for very simple systems.
One very important use of the wavefunction comes from the Born interpretation, which derives information about the location of a particle from its wavefunction.
Quite simply, it states that the square modulus of the wavefunction, |Ψ|2 , at any given point is proportional to the probability of finding the particle at that point. (The quantity |ψ|2 is thus a probability density.)
This is simply illustrated in a 1-dimensional situation, when we can state that if the wavefunction of a particle has a value ψ at the point x, then the probability of finding the particle between x and x + dx is proportional to |ψ|2 dx . Thus the probability of finding the particle between two points a and b is proportional to the integral of the square modulus of the wavefunction, evaluated between limits of a and b:
The square modulus of a wavefunction (or indeed any function in general) is given by ψ ψ*, i.e. by the function multiplied by its complex conjugate. This means that |ψ|2 must always be real and positive.
Since physical properties are directly related only to the square modulus of the wavefunction, it follows that we need not be concerned about the effect on these properties of the wavefunction being complex or negative at a particular point in space. (Note however that the sign of a wavefunction can indirectly be of great significance, as it gives rise to the possibilities of constructive or destructive overlap between different wavefunctions, which is for example crucial in theories of molecular bonding.)
It may be shown that if a wavefunction is an accurate description of a system, then it can be multiplied by any constant factor and still remain an accurate wavefunction for the system. i.e. if Ψ is a wavefunction, then so is NΨ, where N is a constant.
(This is because any wavefunction must satisfy a particular equation known as the Schrodinger equation. The wavefunction occurs on both sides of the equation, so any constant factor may be cancelled. This is covered in more detail under a discussion of the Schrodinger equation, here.)
This allows us to find a constant factor, N, called a normalisation constant, that makes the constant of proportionality in the Born interpretation of the wavefunction equal to one. i.e. it makes the proportionality into an equality.
Now, for a wavefunction NΨ, the probability of finding the particle anywhere in space is proportional to the integral of the square modulus of the wavefunction, N2|Ψ|2 , over all space. However, we also know that the probability of finding the particle somewhere in space must be equal to one (it is certain the particle must be somewhere in space), so we may write:
Evaluation of the integral allows us to calculate the constant N required to normalise the wavefunction. Note that the integral is over all the space that the electron may occupy, eg from + ∞ to -∞ .
From here on, unless otherwise specified, it should be assumed that the wavefunctions used are normalised. i.e. ψ includes a numerical factor that means in one dimension:
In three dimensions, the requirement for a wavefunction to be normalised is:
where dτ is shorthand for dxdydz.