Momentum from position state vector

[nobody101]

Hi all, I'm a newbie to this forum and as a high school student, I only have a basic understanding of quantum mechanics, but here's something that I really want to know.

My question is, if I know the state vector of a quantum particle in the position basis, how do I transform it to the momentum basis? From what I've read, it should involve the Fourier transform, and since I'm interested in (discrete, finite) state vectors rather than continuous wavefunctions, I think I would need the discrete Fourier transform matrix, which I'm familiar with. Also, if I know the state vector that specifies the probability that a particle will be in any given position out of a certain number of positions, can I determine the possible values of the momentum of that particle (that is, the spectrum of eigenvalues of the momentum operator)? On that note, how do I determine the momentum operator of a particle, as a matrix? (since observables of a quantum system correspond to operators, and their eigenvalues give the allowed measurable quantities)

Here's an example of a question I'd like to be able to answer (very contrived):

Say we have a particle, in two dimensions, confined to a two-dimensional box (or square). The square is divided into four equal quadrants. Each quadrant corresponds to a vector in the position basis. The probabilities that the particle will be found in quadrants 1, 2, 3, or 4 are, respectively, 1/3, 1/6, 1/4, and 1/4. Let's say that the state vector for the particle (which tells us about its possible positions) is

$$[\frac{1}{\sqrt{3}}e^{i\theta_1}, \frac{1}{\sqrt{6}}e^{i\theta_2}, \frac{1}{2}e^{i\theta_3}, \frac{1}{2}e^{i\theta_4}].$$

How do I determine the possible values for momenta that the particle can have, and more specifically, the matrix operator corresponding to momentum?


Also, if you know of any books that treat this topic relatively simply, but clearly, I'd like to know of them. I have never taken a course in quantum mechanics.

Thanks in advance.

 

[eljose79]

Hello and welcome to the forum! It's great to see a high school student with an interest in quantum mechanics.

To answer your question, you are correct that transforming a state vector from the position basis to the momentum basis involves using the Fourier transform. However, since you are dealing with discrete, finite state vectors, you would need to use the discrete Fourier transform matrix, as you mentioned.

To determine the possible values of momentum for a given state vector, you can use the momentum operator, which is defined as the derivative of the position operator with respect to time. In matrix form, it can be represented as the derivative of the position matrix. This operator will have a spectrum of eigenvalues, which correspond to the possible values of momentum for that state vector.

In your example, the state vector you provided corresponds to a particle that is equally likely to be found in any of the four quadrants. In this case, the momentum operator would be a diagonal matrix with equal values along the diagonal, representing the fact that the particle has equal probabilities of having any momentum.

As for resources, I would recommend starting with introductory quantum mechanics textbooks such as "Introduction to Quantum Mechanics" by David J. Griffiths or "Quantum Mechanics: Concepts and Applications" by Nouredine Zettili. These books cover the basics of quantum mechanics in a clear and easy-to-understand manner.

I hope this helps answer your question. Keep exploring and learning about quantum mechanics, it's a fascinating field!

 

[denian]

Hi there, thank you for your question. You are correct in thinking that the transformation from the position basis to the momentum basis involves the Fourier transform. The discrete Fourier transform matrix is indeed the one you will need to use for discrete, finite state vectors.

To determine the possible values of momentum for a given state vector, you can use the momentum operator. This operator is usually represented as a matrix, and its eigenvalues will give you the allowed measurable values of momentum for that state vector. The momentum operator is given by the formula p = -iħ(d/dx) in one dimension, and in two dimensions, it takes the form of a matrix with elements -iħ(d/dx) and -iħ(d/dy) on the diagonal. You can find more information about the momentum operator in any introductory quantum mechanics textbook.

In your example, the possible values for momentum will depend on the specific values of θ in your state vector. To determine the exact values, you can use the momentum operator matrix and apply it to your state vector, and then find the eigenvalues of the resulting matrix. These eigenvalues will give you the possible values of momentum for that state vector.

As for books, I would recommend "Introduction to Quantum Mechanics" by David J. Griffiths or "Quantum Mechanics: The Theoretical Minimum" by Leonard Susskind and Art Friedman. Both of these books provide a clear and relatively simple introduction to quantum mechanics, including the concepts of state vectors and operators.

I hope this helps! Keep exploring and learning about quantum mechanics, it's a fascinating subject. Best of luck in your studies.