Quantum Mechanics. Normalised basis wave functions and Eigenvalues.

N} \sum_{i=1}^{N} p_i, we can see that as N (the number of measurements) approaches infinity, the average measurement outcome will approach the expectation value we just calculated, -i \hbar 2 \pi n. This means that the measurement outcomes will be distributed around this value, with a higher probability of getting values closer to -i \hbar 2 \pi n.I hope this helps you understand the concept better. Let me know if you have any further questions. In summary, we looked at the expression for the normalized basis wave functions, found the eigenvalues of momentum, calculated the expectation value of
  • #1
leoflindall
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Homework Statement



Consider a particle with periodic boundary conditions of length L. Write dwon the expression for the normalised basis wave functions and their eigenvalues. Find the eigen value of the momentum and the expectation value of the momentum with respect to one of the momentum with respect to one of these wave functions. Explain the result in terms of measurement outcomes of the momentum.


Homework Equations





The Attempt at a Solution



I haven't had any formal Linear algebre training so I always struggle with these questions.

I know that the wave function for periodic boundary conditions is;

[tex]\Psi[/tex] (x,t) = [tex]\frac{1}{\sqrt{L}}[/tex] eikx - iEt/h

Where h = h bar


and that the expetation value is;

<p> = [tex]\int[/tex] ([tex]\Psi[/tex]* [tex]\hat{p}[/tex] [tex]\Psi[/tex] .dx

where p = -ih d/dx.


and that the measurement outcome is;

<[tex]\hat{p}[/tex]> = lim n[tex]\rightarrow[/tex][tex]\infty[/tex] [tex]\frac{1}{N}[/tex] [tex]\sum[/tex] pi

What I don't understand is how to find the eigen values of momentum from these, and how to explain the result in terms of the measurement outcome.

Can anybody help me understand this?

Many Thanks
 
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  • #2
,




Thank you for your question. I can help you understand how to find the eigen values of momentum and explain the result in terms of measurement outcomes.

First, let's start with the expression for the normalized basis wave functions. For a particle with periodic boundary conditions of length L, the normalized basis wave functions are given by:

\Psi_n(x) = \frac{1}{\sqrt{L}} e^{i \frac{2 \pi n}{L} x}

where n is an integer representing the energy level of the particle. These wave functions satisfy the periodic boundary condition, which means that \Psi_n(x+L) = \Psi_n(x).

Now, to find the eigenvalues of momentum, we can use the momentum operator \hat{p} = -i \hbar \frac{d}{dx} on the wave function \Psi_n(x). This gives us:

\hat{p} \Psi_n(x) = -i \hbar \frac{d}{dx} \Psi_n(x) = -i \hbar \frac{2 \pi n}{L} \Psi_n(x)

So, the eigenvalue of momentum for the wave function \Psi_n(x) is given by -i \hbar \frac{2 \pi n}{L}.

Next, to find the expectation value of momentum, we can use the formula:

<p> = \int \Psi_n^*(x) \hat{p} \Psi_n(x) dx

Substituting the expression for \hat{p} and \Psi_n(x), we get:

<p> = \int \frac{1}{L} e^{-i \frac{2 \pi n}{L} x} (-i \hbar \frac{2 \pi n}{L}) \frac{1}{\sqrt{L}} e^{i \frac{2 \pi n}{L} x} dx

Simplifying this, we get:

<p> = \frac{\hbar}{L} \int_{0}^{L} (-2 \pi n) dx = -i \hbar 2 \pi n

So, the expectation value of momentum for the wave function \Psi_n(x) is -i \hbar 2 \pi n.

Now, the measurement outcome of momentum can be explained in terms of the expectation value we just calculated. Since the measurement
 

1. What is quantum mechanics?

Quantum mechanics is a branch of physics that studies the behavior of matter and energy at the atomic and subatomic levels. It describes the fundamental principles that govern the behavior of particles such as electrons and photons, and how they interact with each other.

2. What are normalised basis wave functions?

Normalised basis wave functions are mathematical representations of the possible states of a quantum system. They describe the probability of finding a particle in a specific location or with a certain energy level. These functions are normalised to ensure that the total probability of finding a particle is equal to 1.

3. What is the significance of eigenvalues in quantum mechanics?

Eigenvalues are important in quantum mechanics as they represent the possible values of a physical observable, such as energy or position. The corresponding eigenfunctions represent the states that the system can exist in, and the eigenvalues determine the probability of measuring a specific value for that observable.

4. How are eigenvalues and normalised basis wave functions related?

Eigenvalues and normalised basis wave functions are closely related as the eigenvalues are determined by the normalised basis wave functions. The eigenvalues correspond to the energy levels of the system, and the normalised basis wave functions describe the probability of finding the particle in a particular energy state.

5. Why is understanding quantum mechanics important?

Understanding quantum mechanics is essential for advancing our understanding of the fundamental laws of nature and developing new technologies. It has practical applications in fields such as electronics, materials science, and cryptography. Additionally, quantum mechanics has revolutionized our understanding of the universe at the smallest scales and has led to groundbreaking discoveries in physics.

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