# B How to Derive Wave Function

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1. Sep 3, 2016

### 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 Schrodinger Equation is derived so easily using this Wave Function. I think it is necessary to understand how the Wave Function is derived.

2. Sep 3, 2016

### Orodruin

Staff Emeritus
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.

3. Sep 3, 2016

### Nipuna Weerasekara

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

4. Sep 3, 2016

### weirdoguy

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

5. Sep 3, 2016

### 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.

6. Sep 3, 2016

### Mentz114

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

7. Sep 3, 2016

### Staff: Mentor

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. )

Last edited: Sep 3, 2016
8. Sep 3, 2016

### kent davidge

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.