Undergrad What is the Wave Function and How Does it Compare to the Schrodinger Equation?

[Xilus]

What is the wave function? does it have several different forms?
how does the Schrödinger equation compare to the wave function?

 

[michael879]

What is the wave function? does it have several different forms?
how does the Schrödinger equation compare to the wave function?

The wave function is a mathematical construct that has no (or many) clear physical meaning. The Schrödinger equation is one of a handful of equations (Dirac and Klein-Gordon are the main alternative single particle equations) governing the evolution of this wave in space and time. The math behind quantum mechanics and its physical predictions are well defined, but interpreting what it means is much more complicated. The wave function itself is not a physical observable, meaning you can never measure or see it. This has given way to a huge number of interpretations of quantum mechanics, which attempt to give a philosophical meaning to the math. The two most common are:

Copenhagen interpretation, which treats the wave function as a 'real' object that collapses into a classical value under measurement. This is basically just a literal interpretation of the underlying equations
Many-worlds interpretation, which treats every possible classical world allowed by the wave function as a distinct universe. Here, measurement merely splits the universe around the observer, producing multiple observers which each measure something different.

These are just two though. There are an endless number of interpretations which, by definition, predict the exact same outcomes to any experiment and are therefore indistinguishable

 

[Xilus]

Thanks for the response. Can you explain more of the mathematics of the wave function?
Is this it?

 

[michael879]

This is the 1-dimensional time-independent Schrödinger equation for a free particle. So by using this equation, as opposed to the general one, you're making some assumptions:
1) 1-dimensional: this particle is confined to 1 spatial dimension
2) time-independent: this particle has a fixed energy (i.e. it is an eigenstate of the Hamiltonian)
3) free: this particle is not under any external forces, which would produce a potential energy term
Given the restrictive nature of this equation, the solution can be easily expressed as
\Psi(x) = Ae^{ikx} + Be^{-ikx}
where A and B must be determined by boundary and normalization conditions.

As I mentioned above, the wave function isn't a physical observable. However, its absolute square |\Psi|^2 is, and represents the probability density for finding a particle at a point x. To calculate the probability of the particle being observed between two points a and b, you just need to integrate:
P(a,b) = \int^a_b |\Psi(x)|^2dx