Sample Space for Free Particle in the general case

In summary, the conversation discusses the solution to Schrodinger's Equation for a free particle in the general case. It is noted that the given solution is not normalizable and does not accurately represent a free particle, but can still be useful for certain applications. The idea of using a wave packet instead is also mentioned.
  • #1
IronHamster
28
0
I am a beginner to quantum mechanics and am trying to make sense of Schrodinger's Equation. I am attempting to find probabilities in the case of a free particle in the general case.

It is my understanding that the solution to Schrodinger's Equation in the general case of a free particle is as follows:

[tex]\psi(X,T) = e^{i/\hslash ( px - Et)}[/tex]

The modulus square of this is 1, which means the probability density function is uniform.

Two questions:
1. Over what values of x is this pdf defined? Can we eliminate all values of x > ct?
2. Am I correct to interpret x as the distance from the (known) starting position of the particle at t = 0?

Thanks.
 
Physics news on Phys.org
  • #2
Notice that that wave-function is not normalizable. The integral of the modulus square of that wave-function over all space is infinite. A free particle wave function cannot actually be what you gave, but must be a wave-packet.
 
  • #3
Matterwave said:
Notice that that wave-function is not normalizable. The integral of the modulus square of that wave-function over all space is infinite. A free particle wave function cannot actually be what you gave, but must be a wave-packet.

So are you saying that the solution I mentioned does not describe a free particle wave? I'm not sure how that could be, I have read from multiple sources that it is.

Is there a different approach that needs to be taken to achieve a normalizable function?
 
  • #4
A real free particle cannot be represented by that function because that function is not normalizable. A real free particle is represented by a wave packet. That function is a function of a particle with exactly 1 momentum (p), but really a particle is represented by a wave packet which has a range of momenta.

You can say that your equation is only a "partial" solution. It hasn't been fixed up yet.

Still, that function is useful for many applications. For example, if we are doing a scattering problem off of a finite square barrier, we tend to just use that function. The central results you obtain by using that function (the transmission and reflection coefficients) is surprisingly good to the results you would get if you made wave packets; however, doing a scattering problem with wave packets is a nightmare.
 
  • #5
Oh ok that makes sense. Thanks!
 

What is a sample space for a free particle in the general case?

A sample space for a free particle in the general case is a mathematical concept used to describe all possible outcomes or states that a particle can have in a given system. It includes the position, momentum, and energy of the particle, and is often represented as a multi-dimensional space.

How is a sample space for a free particle in the general case different from a sample space for a confined particle?

A sample space for a confined particle is limited to a specific region in space, while a sample space for a free particle in the general case includes all possible positions and momenta that the particle can have in a larger, unconfined system. Additionally, a confined particle may have certain restrictions or boundaries on its movement, while a free particle has no such limitations.

Can a sample space for a free particle in the general case be visualized?

Yes, a sample space for a free particle in the general case can be visualized as a multi-dimensional graph, with each axis representing a different aspect of the particle's state. For example, one axis could represent position and another could represent momentum. This visual representation can help in understanding the behavior of a free particle in a given system.

How is a sample space for a free particle in the general case related to quantum mechanics?

In quantum mechanics, the sample space for a free particle in the general case is described by the wave function, which represents the probability of finding the particle in a particular state. The square of the wave function gives the probability density of finding the particle at a specific position in space. The behavior of a free particle in the general case is also governed by the Schrödinger equation, which describes the time evolution of the wave function.

What factors can affect the sample space for a free particle in the general case?

The sample space for a free particle in the general case can be affected by various factors such as the strength and direction of external forces, the presence of other particles or objects in the system, and the shape and boundaries of the system itself. These factors can influence the position, momentum, and energy of the particle, thus altering its sample space.

Similar threads

Replies
25
Views
1K
Replies
4
Views
835
Replies
1
Views
740
Replies
17
Views
1K
  • Quantum Physics
Replies
27
Views
2K
  • Quantum Physics
Replies
23
Views
577
Replies
7
Views
2K
Replies
1
Views
851
  • Quantum Physics
Replies
4
Views
1K
Replies
1
Views
737
Back
Top