Wave functions and probability densities

In summary, the wave function is a function of all the positions and time of a particle, and can be found by solving Schrödinger's equation. The wave function for a particle in a box can be found by solving the equation for the energy of the particle as a function of x. The U and E in Schrödinger's equation are measured quantities.
  • #1
jaredogden
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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
 
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  • #2
jaredogden said:
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.
 
  • #3
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.
 
  • #4
jaredogden said:
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).
 
  • #5
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
 
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What is a wave function?

A wave function is a mathematical function that describes the behavior and properties of a quantum system. It is used to calculate the probability of finding a particle at a certain position or state.

What is a probability density?

A probability density is a function that describes the likelihood of a particle being found at a particular position in space. It is represented by the square of the wave function and is used to determine the probability of finding a particle within a given region.

How are wave functions and probability densities related?

Wave functions and probability densities are closely related. The square of the wave function represents the probability density, which describes the likelihood of finding a particle at a specific position. In other words, the wave function gives the amplitude of the particle's wave, while the probability density gives the intensity of the wave at a particular point.

What is the uncertainty principle?

The uncertainty principle states that it is impossible to know both the exact position and momentum of a particle at the same time. This is due to the wave-like nature of particles, which means that their position and momentum cannot be measured simultaneously with 100% accuracy. The uncertainty principle is a fundamental principle in quantum mechanics and has important implications for our understanding of the physical world.

What is the role of wave functions and probability densities in quantum mechanics?

Wave functions and probability densities play a central role in quantum mechanics. They are used to describe the behavior and properties of quantum systems, and are essential for calculating the probabilities of different outcomes in quantum experiments. The wave function also contains information about the energy and momentum of a particle, which can be used to make predictions about its behavior.

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