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Wave function is the mathematical quantity that describes the quantum state of a particle. It talks about the spin, momentum, position, and time of the particle. It talks about the possibility of a given particle at that time and is derived from the variables of Schrodinger’s equation.
In this article, you will learn about wave function, its derivation, properties, significance, and various applications.
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A wave function describes the quantum state of an isolated system. It is denoted by the Greek letter \(\psi\). This is the wave function symbol. The probability of each particle or electron in quantum mechanics is represented by the wave function in a third-dimensional world. We find this by including an imaginary number which is squared to find a real solution representing the position of a particle or an electron.
We obtain the wave function by Schrodinger’s equation :
1) Time dependent: \(ih\dfrac{\delta \psi}{\delta t}=\dfrac{-h^2}{2m}\dfrac{\delta^2\psi}{\delta x^2}+V(x)\psi(x,t)\) and
2) Time independent: \(E \psi (x)= \dfrac{-h^2}{2m}\dfrac{\delta^2\psi}{\delta x^2}+U(x)\psi (x)\)
Let’s understand the wave function derivation.
Bohr predicted the energies of an electron, it relied on classical principles and couldn’t explain the quantum phenomena of an atom.
According to Maxwell’s equations, every part of a magnetic field or an electric field always has a solution in a 3D space. Hence the wave equation in a 3D space can be obtained as follows:
The amplitude of a wave at any time t is the function of its displacement x and the equation for the wave motion can be formulated as,
\(\dfrac{\delta^2y}{\delta x^2}=\dfrac{1}{v^2}\dfrac{\delta^2y}{\delta t^2}\)
Y is a function of x and t,
\(y=f(x)f'(t)\)
Here f(x) and f’(t) are functions of x and t respectively,
Consider a standing wave created in a string which is fixed between two points, stroke from one side would generate a positive amplitude and its reflection from the other end would generate and negative amplitude, in the form of a sine wave, by changing the vibrational frequency we get the fundamental node, first overtone, second overtone, third overtone etc.
The mathematical description of such wave would be,
\(y=f(x)=Asin2\pi\omega t\)
\(\omega\) is the frequency of vibration
Differentiating the above equation with respect to (t),
\(\dfrac{\delta^2y}{\delta t^2}==4\pi^2v^2f(x)f'(t)\)
And now differentiating with respect to x,
\(\dfrac{\delta^2y}{\delta x^2}=f'(t)\dfrac{\delta^2 f(x)}{\delta x^2}\)
\(\dfrac{\delta^2 f(x)}{\delta^2 x}=\dfrac{-4\pi^2v^2}{v^2}\)
The function f(x) shall be replaced with
\(\psi(x)\)
\(\dfrac{\psi^2(x)}{\delta x^2}+\dfrac{4\pi^2m^2v^2}{h^2}\psi=0\)
Total energy (E) is the sum of kinetic energy and potential energy,
\(E=\dfrac{mv^2}{2}+V\ \)
\(\dfrac{\delta^2\psi}{\delta x^2}+\dfrac{\delta^2\psi}{\delta y^2}\dfrac{\delta^2\psi}{\delta z^2}-\dfrac{8\pi^2m}{h^2}(E-V)\psi=0\)
The equation mentioned above is the most popular form of Schrodinger’s equation and is also known as the wave equation.
Also denoted by,
\(\dfrac{-h^2v^2\psi}{2mvx^2}+V(x)\psi= ih \dfrac{v\psi}{vt}\)
The wave function ψ is a central concept in quantum mechanics, representing the state of a quantum system. However, it can have both real and imaginary components, making it challenging to assign a direct physical interpretation.
Max Born proposed an interpretation where the meaningful quantity is |ψ|^2, known as the probability density. According to this interpretation, if ψ is the wave function of a particle within a small volume dv, then |ψ|^2 dV gives the probability of finding the particle in that region at a specific time.
It ensures that, since electrons must exist somewhere in space, a normalized wave function is one that satisfies this condition, ensuring that the total probability of finding the particle is equal to 1.
\(\oint_v |\psi|^2 dv = 1\)
The following pointers discusses the significance of wave function.
Example 1. Consider a particle of mass m with zero energy has a time-independent wave function, \(\psi(x)= Axe^{\dfrac{-x^2}{L^2}}\). Determine the potential energy of the particle.
Solution.
Schrodinger’s equation for time-independent wave function is given as,
\(\dfrac{-h^2}{2m}\dfrac{d^2\psi x}{dx^2}+U(x)\psi (x)= E\psi(x)\) ……….eq (1)
On substituting \(\psi(x)= Axe^{\dfrac{-x^2}{L^2}}\) in ………..eq (1)
For a particle with zero energy,
\(U(x)= \dfrac{2h^2}{mL^4}\ (x^2-\dfrac{3L^2}{2})\)
And U(x) is the parabola with x = 0,
\(U(0)=\dfrac{-3h^2}{mL^2}\)
Which is the potential energy of the particle.
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If you are checking Wave Function, also check the related physics articles: | |
Derivation of Schrodinger Wave Equation | Quantum Mechanics |
Helmholtz Equation | Dirac Equation |
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