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# Which of these are the characteristics of the wave function ${{\Psi }}$ ?A.${{\Psi }}$ must be single-valuedB.$\int\limits_{{{ - }}\infty }^{{{ + }}\infty } {{{{\Psi }}^{{2}}}} {{dxdydz = 1}}$C.${{\Psi }}$ must be finite and continuousD.${{\Psi }}$ represents a standing wave

Last updated date: 19th Sep 2024
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Hint: A wave function is a mathematical expression. It is defined as a mathematical description of the quantum state of an isolated quantum system in quantum physics. The wave function associated with a moving particle is not an observable quantity and does not have any physical meaning. It is a complex quantity and represents the probability density of the possibility of finding an electron in a particular region in a given instant of time.

A wave function can be obtained by using an imaginary number that is squared to get a real number solution which results in the position of the particle. Some of the properties of the wave function ${{\Psi }}$ are as follows:
All measurable information about the particle is available in a wave function.
${{\Psi }}$ should be continuous and single-valued.
Energy calculations are using the Schrodinger wave equation.
Probability distribution in three dimensions is established using the wave function.
The Square of the wave function ${{{\Psi }}^{{2}}}$ gives the probability of finding a particle at a particular place in the given instant of time.
The probability of finding a particle is 1 if it exists.

Looking at all the above properties, we can say that the correct answer is option A.

Note:
The concept of wave function was first introduced by Erwin Schrodinger in the year 1925 with the help of the Schrodinger equation. The Schrodinger equation can be defined as a linear partial differential equation describing the wave function, ${{\Psi }}$. Schrodinger equation is:
$[\dfrac{{ - {h^2}}}{{2m}}{\nabla ^2} + V(r)]\Psi (r) = E\Psi (r)$ , it is a time-independent equation.
M is the Mass of the particle
$\nabla$ is Laplacian operator
h is Planck constant
E is constant which is equal to the energy level of the system