Quantum mechanics (from quantum waves S1)

Equations, theory etc... for the quantum mechanics element of classical and quantum waves.

What is the space-time version of the wave function

Ψ(x,t) = Ae^i(kx-wt)

What is the space-only version of the wave function

𝛙(x) = Ae^ikx

What is Born’s interpretation (of the wave function)?

where dx is a small change in x. (delta x notation can also be used for dx).

* indicates that we use the complex conjugate (of the wave function). Not a multiplication.

He related the absolute square of space-only wave function to a probability density. This is a postulate.

What is the absolute square of the wave function equal to?

probability density

Properties of the wave function

• It must be single-valued and continuous

• It’s first derivative must be continuous **(why??)

—> This condition can be dropped at infinity potential boundaries

• It must be normalisable

Postulates of quantum mechanics (=Bohr’s postulates??)

• The properties of any quantum mechanical system are contained within the wavefunction

• The wavefunction is interpreted as the probability amplitude with the absolute square interpreted as the probability density

How do we normalise the wave function?

see image for equation

Why must we normalise the wave function?

To give the wave function a physical meaning. It gives the probability density. When we integrate this, it gives the probability of an event happening within a given range, which = 1 from - infinity to + infinity.