Wave functions and probability densities

[jaredogden]

I am reading over some quantum mechanics and have came across wave functions and probability densities. Needless to say I am Havin difficulties understanding exactly what they are. If anyone can help me understand what exactly they are and just any information please post it. Thanks

 

[SpectraCat]

I am reading over some quantum mechanics and have came across wave functions and probability densities. Needless to say I am Havin difficulties understanding exactly what they are. If anyone can help me understand what exactly they are and just any information please post it. Thanks

I can help you understand what the definitions of those things are within the context of standard quantum mechanics, however you should know that on this forum, there is active debate about the true physical significance of both things.

The wavefunction is defined by one of the postulates of quantum mechanics (typically numbered as postulate 1). The wavefunction depends on all of the positions of all particles making up a system, and also depends on time. The wavefunction describes the state of the quantum system, in that it allows one to calculate or predict (within certain limits) all of the measureable physical properties of the system. Not all measurements on a quantum system give a predictable result however .. the Heisenberg Uncertainty Principle tells us that certain properties (e.g. position and momentum) cannot both simultaneously be known to arbitrary precision.

The probability density is defined by another postulate of quantum mechanics, known as the Born interpretation (because it is attributed to Max Born). It says that the square modulus of the wavefunction is proportional to the probability density for observing the system at a given set of coordinates. This is in contrast to the wavefunction itself, which is interpreted as a probability amplitude in the same context, and has no direct physical meaning. The probability density is always real and non-negative, but the wavefunction itself is in general a complex quantity in the mathematical sense (i.e. it has components that are mathematically both real and imaginary).

I don't know if this just repeats what you have already read, but I hope it helps at least a little bit.

 

[jaredogden]

That actually did help. It gave a new wording to what I already read and sometimes that is all you need.

I have another question if you or anyone else can answer, I'm not a physics major but an ME so I'm not real real sharp with quantum. However if anyone can help explain, how is it that you would find a wave function from a system? It seems that the function would be so complex, however there is given wave functions for electrons in a box and hydrogen atoms and such. Are these just found through graphical interpretation of experiments and taken from the line that best represents data points?

I'm not sure if I'm even close but I just would like to understand this stuff even more it's intriguing to me.

 

[jtbell]

there is given wave functions for electrons in a box and hydrogen atoms

For these two examples, and some other simple situations, the wave functions can be found by solving Schrödinger's equation (the differential equation that the wave function satisfies).

 

[jaredogden]

So then take the wave function for a particle in a box for example being (forgive me for not using symbols I am on my phone) psi(x)=Asin(2(pi)x/lambda) the wave function is a function of x because it is a one dimensional problem. Since it is not a function of time we can use schrodingers time-independent equation with the values of m for an electron and the other known
constants to solve for psi(x)
correct? Would the U and E in schrodingers equation be measured quantities or where would they come from? Also from what function would you be taking the second derivative of psi with respect to x from in schrodingers equation?

Hopfully that made sense, maybe I should get on a computer so I can type better haha.

EDIT: I think I figured it out.. solve for d^2(psi)/dx^2 in schrodingers equation then take the second derivative correct?
And U is a constant being the
boxs height so it is 0 after a derivative is taken?

Double edit: wrong again so U=0 then solve for d^2(psi)/dx^2 thy IS the second derivative, you don't take the second derivative. Man my calc is rusty I guess haha. I'm sure no one understands what I'm saying, I'm just talking it out I guess. Don't judge me! Haha