Wave Function

A wave function, in quantum physics, refers to a mathematical description of a particle’s quantum state as a function of spin, time, momentum, and position. Moreover, it is a function of the degrees of freedom that correspond to a maximal set of commuting observables. Furthermore, psi, 𝚿, is the wave function symbol.

Introduction to Wave Function

With wave function, one can explain the probability of finding, within the matter wave, an electron. Furthermore, one can obtain it by involving an imaginary number whose squaring takes place to find a real number solution. Moreover, this real number solution would result in an electron’s position.

In three dimensions, the wave function would establish the probability distribution. Furthermore, with a wave function, 1 is the probability of finding a particle.

How do we Measure Wave Function?

The measurement of the wave function can take place with the help of the Schrodinger equation. Furthermore, experts define the Schrodinger equation as the linear partial differential equation that can describe the wave function, 𝚿.

The name of this equation is after Erwin Schrodinger. Schrodinger could work on the wave function by making use of the postulates of quantum mechanics.

Formula of Wave Function

For the wave function formula, one must look to the Schrodinger equation. Below is the Schrodinger equation:

Time dependent Schrodinger equation:  \(ih\frac{\vartheta }{\vartheta t}\Psi \left ( r,t \right )\) = \(\left [ \frac{h^{2}}{2m}\triangledown ^{2}+V\left ( r,t \right )  \right ]\Psi \left ( r,t \right )\)

Time independent Schrodinger equation: \(\left [ \frac{-h^{2}}{2m}\triangledown ^{2}+V\left ( r \right ) \right ]\Psi \left ( r \right )\) = \(E\Psi \left ( r \right )\)

Where,

m: mass of the particle

i: imaginary unit

h=h/2𝝿: Planck constant that is reduced

E: constant that is equal to the energy level of the system

Derivation of the Formula of Wave Function

An observation was made by the experts that one can write the wave function of a particle of fixed energy E as a linear combination of wave functions of the form

\(\Psi \left ( x,t \right ) = Ae^{i\left ( kx-wt \right )}\)          (1)

This is representative of a wave that travels in the positive x direction, and a corresponding wave that travels in the opposite direction. Therefore, the result is a standing wave which is in accordance with the boundary conditions.

This corresponds, in an intuitive manner, to the classical notion of a particle bouncing. Furthermore, this particle bounces back and forth between the potential well walls. Most noteworthy, it is possible to adopt the wave function as being the suitable wave function for a particle that is free.

The momentum of this free particle is p = hk and its energy is E = hω. So, one can see that

\(\frac{\vartheta ^{2}\Psi }{\vartheta x^{2}} = -k^{2}\Psi\)          (2)

By making use of E = p²/2m = h²k²/2m, one can write

\(\frac{-h^{2}\vartheta ^{2}\Psi }{2m\vartheta x^{2}} = \frac{p^{2}\Psi }{2m}\)           (3)

Furthermore, similarly

\(\frac{\vartheta \Psi }{\vartheta t} = -i\omega \Psi\)            (4)

One can also write this, using E = hω:

\(ih\frac{\vartheta \Psi }{\vartheta t}\) = hωψ = \(E\Psi\)                     (5)

Now generalizing that this is possible to the situation in which there is a potential energy as well as a kinetic energy present, then E = p²/2m + V (x) so that

\(E\Psi = \frac{P^{2}\Psi }{2m} + V(x)\Psi\)                (6)

where \(\Psi\) is now the wave function of a particle whose movement takes place in the presence of a potential V (x). However, considering the results Eq. (6.3) and Eq. (6.5) still apply in this case, then what we are left with is

\(-\frac{h^{2}\vartheta ^{2}\psi }{2m\vartheta x^{2}} + V\left ( x \right )\Psi = ih\frac{\vartheta \psi }{\vartheta t}\)                 (7)

Most noteworthy, this is the Schrödinger wave equation.

FAQs For Wave Function

Question 1: What is meant by wave function?

Answer 1: A wave function is a mathematical description of the quantum state of any quantum system that is isolated. Moreover, it is a complex-valued probability amplitude.

Question 2: Does the wave function has any physical significance?

Answer 2: The wave function ψ is a complex quantity. Furthermore, this function is not an observable quantity when it has an association with a moving particle. Also, it lacks any direct physical meaning.