How is the Wave Function Derived in Quantum Mechanics?

[Nipuna Weerasekara]

Everybody knows what is the Wave Function is.
$$\Psi=\space e^{i(kx-\omega t)}$$
or
$$\Psi=\space cos{(kx-\omega t)} \space - \space isin{(kx-\omega t)}$$

But can anyone tell me how it is derived. Since Schrödinger Equation is derived so easily using this Wave Function. I think it is necessary to understand how the Wave Function is derived.

 

[Orodruin]

Everybody knows what is the Wave Function is.
$$\Psi=\space e^{i(kx-\omega t)}$$
or
$$\Psi=\space cos{(kx-\omega t)} \space - \space isin{(kx-\omega t)}$$

The wave function of what? You have just given the wave function for a free particle. In many other cases the wave function will be completely different and behave different wrt time.

 

[Nipuna Weerasekara]

This is the wave function I found when proving the Schrödinger equation. I think this is the wave function of the electron in Hydrogen atom.

 

[weirdoguy]

I think this is the wave function of the electron in Hydrogen atom.

No, it's definitely not a wave function of electron in hydrogen. You need to read more carefully your sources.

 

[Nipuna Weerasekara]

Alright, let us think this is a Wave function of a particle for the time being, can you tell me how it is derived.

 

[Mentz114]

Alright, let us think this is a Wave function of a particle for the time being, can you tell me how it is derived.

Classically, it is the solution to the EOM of the simple harmonic oscillator.

 

[jtbell]

Since Schrödinger Equation is derived so easily using this Wave Function.

It's the other way around. One derives the wave function for a particular system (e.g. a free particle) by solving Schrödinger's equation for that system.

Many introductory textbooks "justify" the Schrödinger equation or "motivate" it or "make it plausible" by assuming that a free particle with a definite momentum must be represented by a simple harmonic wave $\Psi(x,t) = Ae^{i(kx - \omega t)}$, but fundamentally, the Schrödinger equation comes first. Many or most elementary treaments of QM simply present the SE as a fundamental assumption of the theory.

(I predict that we will now have a long debate about what is the "real" logical starting point for non-relativistic quantum mechanics. )

 

[kent davidge]

Many introductory textbooks "justify" the Schrödinger equation or "motivate" it or "make it plausible" by assuming that a free particle with a definite momentum must be represented by a simple harmonic wave Ψ(x,t)=Aei(kx−ωt)

(I predict that we will now have a long debate about what is the "real" logical starting point for non-relativistic quantum mechanics. )

I think the best derivation is founded on Universty Physics With Modern Physics by Sears and Zemansky. I did not find any derivation even in more advanced textbooks on QM, because they usually apresents the wave function only as an postulate.