Question

A) Where does the wavefunction have any amplitude at all? -- conservation of energy

B) What is the frequency (density of nodes) at different locations? -- kinetic energy

C) What is the amplitude (hence amplitude of Ψ ∗ Ψ ) -- probability density

Answer 1

A) Amplitude of wave function does not have any physical meaning.Its modulus square gives the probability for finding the particle at a particular point as a function of x.It is related to conservation of energy as energy must be conserved which is related to how energy will be distributed across as a function of distance.Again amplitude should be such that the total probability of finding the particle over all space is unity.

B)As is apparent form Schrodinger equation second order derivative of the wave function is proportional to the Energy of the particle.More number of nodes mean that curvature is higher hence higher frequency which implies greater amount of kinetic energy.

C)Probability density of the particle is defined as the modulus square of the wave function as a function of x.It is positive definite and normalised to unity.For non relativistic cases wave function is usually the solution of the schordinger equation which is a second order partial differential  equation.