Probabilistic interpretation of wave function

[coki2000]

I just start to study quantum mechanics and i don't understand why square of wave function is probability density function. I think the reason of taking square of wave function is because we should eliminate complex compounds of wave function to get a real magnitute. But after why don't we take square root of that? Is there any mathematical proof which explains the reason? Thanks for your help.

 

[Bill_K]

coki2000, This is the most basic, fundamental axiom of quantum mechanics. You are not allowed to ask why! Seriously, no one knows why, except the fact that everything depends on a complex probability amplitude and not just a real probability is what makes quantum mechanics so different from classical mechanics. It makes interference possible.

 

[coki2000]

Thank you for your reply.

"The discovery that the symbol $$\Psi$$ in the Schrödinger equation represents a probability density took many years and arose only after much work by many physicists."

I have read this from a book of quantum mechanics so I wonder this "much work" in the quote. It is also mentioned that the Schrödinger equation cannot be
derived from anywhere else and it is just like the F=ma but I can't understand where $$-\frac{\hbar}{2m}$$ and $$i\hbar$$ come from. It is a phenomenological equation but also very sophisticated than F=ma. So I think there must be an explanation to where they come from. Thanks.

 

[Bill_K]

Again, the fact that Ψ(x) represents a probability amplitude is an axiom. You can't ask where an axiom came from. You can't say to Euclid, "Hey, where did you get that parallel postulate?" But what you *can* do is verify how well it works, and see that it does not contradict anything else. That's what the rest of the course will be about.

The discovery that the Ψ(x) in the Schrödinger equation represents a probability density took many years and arose only after much work by many physicists.

I don't agree with that. Ψ(x) always was a probability amplitude. But even if it was not, it is a mistake to try to learn a subject by retracing its history. If you want to learn relativity, the last thing you should do is try to go through Einstein's early papers to see what he was thinking 100 years ago and how he was led to the theory. It's irrelevant, not to mention laborious and time-wasteful. Doing this means you will drag through every mistake he made, every misunderstanding he ever had. We understand the theory now better than Einstein ever did. Learn it the right way from the beginning, not the umpteen wrong ways.

Many blind alleys were followed during the development of quantum mechanics, and many of these incorrect ideas are still around, occupying people's minds. Just realize from the beginning that quantum mechanics is not classical mechanics, and cannot be understood by trying to force it into that mold.

 

[Mike Pemulis]

It is a phenomenological equation but also very sophisticated than F=ma. So I think there must be an explanation to where they come from. Thanks.

There is -- the constants you mentioned are not arbitrary, despite appearances. Do you have a copy of Sakurai's Modern Quantum Mechanics? I would refer you to sections 1.6, 1.7, 2.1, and 2.4 for really good explanations of where the Schrodinger equation comes from. Like Bill_K said, it can't be derived as such, but Sakurai does a really good job of motivating it from a couple of simple principles.

 

[netheril96]

Thank you for your reply.I have read this from a book of quantum mechanics so I wonder this "much work" in the quote. It is also mentioned that the Schrödinger equation cannot be
derived from anywhere else and it is just like the F=ma but I can't understand where $$-\frac{\hbar}{2m}$$ and $$i\hbar$$ come from. It is a phenomenological equation but also very sophisticated than F=ma. So I think there must be an explanation to where they come from. Thanks.

On one of my books it first derives the momentum operator
$$\mathbf{p}=-i\hbar\nabla$$
then constructs the Hamiltonian operator (if you don't know what is Hamiltonian, think of it as energy; most of time it is correct to assume that) from correspondence principle
$$H=\frac{\mathbf{p}^{2}}{2m}+V=-\frac{\hbar^{2}}{2m}\nabla^{2}+V$$
Then it goes on the explain why Hamilton corresponds to time evolution, and concluded that for any allowed wavefunction,
$$\hat{H}\psi=i\hbar\frac{\partial}{\partial t}\psi$$

It is not a real "derivation", but it is easiest to remember Schrodinger equation.