Question

Which of these are the characteristics of wave function $\psi $?
A)$\psi $ Must be single valued.
B)\[\int_{ - \infty }^{ + \infty } {{\psi ^2}dxdydz = 1} \]
C) $\psi $ Must be finite and continuous.
D) $\psi $ Can be discrete.

Answer 1

Hint: We know that Quantum mechanics is a branch of science that deals with the study and behaviour of matter as well as light. The wave function in quantum mechanics can be used to explain the wave properties of a particle. Thus a particle's quantum state can be described using wave function.

Complete step by step answer:
The interpretation of wave function helps us to define the probability of the quantum state of an element as a function of position, momentum, time and spin. It is represented by Greek alphabet psi $\psi $.
Though, it is important to note that there is no physical importance of wave function itself. Nevertheless its proportionate value of ${\psi _2}$ at a given time and point of space does have physical importance.
Let us discuss the properties of wave function.
Properties of Wave Function:
Wave function \[\Psi \] contains all the measurable information about the particles.\[\int_{ - \infty }^{ + \infty } {{\psi ^2}dxdydz = 1} \] This covers all possibilities.
The wave function \[\Psi \] must be finite and continuous because the wave function should be able to describe the behaviour of a particle across all potentials, in any region.
The wave function \[\Psi \] must be single valued. This is a way of guaranteeing that there is only a single value for the probability of the system being in a given state.
We know that $\psi $ is continuous. Therefore, the option D is incorrect.
Hence, all the given options (A), (B) and (C) are correct.

Note:
We must remember that the linear partial equation describing the wave function is called the Schrodinger equation. The equation is named after Schrodinger.
The equations of Schrodinger equation are,
Time dependent Schrodinger equation: \[ih\partial \partial t\Psi \left( {r,t} \right) = [{h^2}2m{\triangledown ^2} + V\left( {r,t} \right)]\Psi \left( {r,t} \right)\].
Time independent Schrodinger equation: \[[ - {h^2}2m{\triangledown ^2} + V\left( r \right)]\Psi \left( r \right) = E\Psi \left( r \right)\].