Normalisation and normalising wavefunctions

[indeterminate]

In our physics class of quantum mechanics, we constantly talk about normalisation and normalising wavefunctions such that the total probability of finding the particle in infinite space is one. I don't get why do we normalise and how do we normalise(I have not taken up statistics course). It would also be benificial if you can provide me with external links.

 

[jtbell]

why do we normalise

If the wave function ψ is normalized, then we can calculate the probability that the particle lies between x=a and x=b (in the one-dimensional case) simply by evaluating the integral $$P(a \leq x \leq b) = \int^b_a {\psi^*\psi dx}$$

and how do we normalise

An un-normalized ψ has an arbitrary constant overall multiplicative factor, call it A. For example, we might have $$\psi(x) = Ae^{-x^2}$$ To normalize ψ, we find the value of A that makes this equation true: $$\int^{+\infty}_{-\infty} {\psi^*\psi dx} = 1$$ That is, we evaluate the integral, whose result must include a factor of A², solve the equation for A, and finally substitute our newly-found value of A back into the original formula for ψ.

 

[indeterminate]

so, basically normalisation is a technique which helps in mathematical calculations of probability. It has no physical significance in the wave equation.

 

[cgk]

so, basically normalisation is a technique which helps in mathematical calculations of probability. It has no physical significance in the wave equation.

As long as you always evaluate expectation values as $$\langle A\rangle = \frac{\langle \Psi|\hat A|\Psi\rangle}{\langle\Psi|\Psi|\rangle},$$
then yes, that is a valid perspective---but not one shared by all instructors, so be careful in examinations. But in any case, note that the wave function property of being normalizable (not normalized) can have important consequences.