Eigenvalues and Probabilities

In summary, a Hermitian operator A with eigenvectors |A1>, |A2>, and |A3> has corresponding eigenvalues a1 = (3)^(-1/2), a2 = 2((3)^(-1/2), and a3 = (5/3)^(1/2). The system's state given by |psi> = ((3)^(-1/2))|A1> + 2((3)^(-1/2))|A2> + ((5/3)^(1/2))|A3> can be normalized to find the probabilities of measuring each eigenvalue, which is equal to the square of the coefficient of the corresponding eigenvector in
  • #1
bluebandit26
8
0

Homework Statement



Suppose that a Hermitian operator A, representing measurable a, has eigenvectors |A1>, |A2>, and |A3> such that A|Ak> = ak|Ak>. The system is at state:

|psi> = ((3)^(-1/2))|A1> + 2((3)^(-1/2))|A2> + ((5/3)^(1/2))|A3>.

Provide the possible measured values of a and corresponding probabilities.


Homework Equations



(A)(psi) = sum[anCnPsin]

The Attempt at a Solution


It would seem that A1 = (3)^(-1/2), A2 = 2((3)^(-1/2), and A3 = (5/3)^(1/2), but the state is not normalized so the probabilities don't add up to one... so I am confused about how to handle this.
 
Physics news on Phys.org
  • #2
So normalise |psi>.
BTW, the probabilities of receiving the eigenvalue a_k is the square of the coefficient of |A_k> in |psi>.
 
  • #3
In general, if [tex]|\psi>=\sum c|n>[/tex] the the probability to find |psi> in the state |n> (and measuring its eigenvalue) is |c|^2=cc* where c* is the complex conjugate of c.
 

1. What are eigenvalues and how are they related to probabilities?

Eigenvalues are a concept in linear algebra that represent the values for which a linear transformation does not change the direction of a vector. In the context of probabilities, eigenvalues are used in the calculation of Markov chains, which model the probability of transitioning between different states.

2. How are eigenvalues and probabilities used in data analysis?

Eigenvalues and probabilities are important in data analysis because they can be used to reduce the dimensionality of large datasets. By identifying the most important eigenvalues and their corresponding eigenvectors, we can summarize the data and make predictions based on the probabilities of transitioning between different states.

3. Can eigenvalues and probabilities be negative?

Yes, eigenvalues and probabilities can be negative. In the context of eigenvalues, a negative eigenvalue indicates that the transformation reverses the direction of the vector. In probabilities, negative values can occur when there is a negative correlation between two variables, meaning that as one increases, the other decreases.

4. What is the relationship between eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are closely related. An eigenvector is a vector that remains in the same direction after a linear transformation, and its corresponding eigenvalue represents the stretching or shrinking factor of the transformation along that vector.

5. How can eigenvalues and probabilities be calculated?

Eigenvalues can be calculated by finding the roots of the characteristic polynomial of a matrix. For probabilities, they can be calculated using the transition matrix of a Markov chain and the power method, which iteratively calculates the dominant eigenvalue and its corresponding eigenvector.

Similar threads

Reproduce band structure Kagome Fermi-Hubbard using Python
    • Advanced Physics Homework Help
Replies
6
Views
130
What To Project Upon When Doing QM Measurements?
    • Advanced Physics Homework Help
Replies
9
Views
205
Probability of finding a particle in the right half of a rectangular potential well
    • Advanced Physics Homework Help
Replies
16
Views
523
Two-level Quantum System - initial state
    • Advanced Physics Homework Help
Replies
14
Views
1K
Switching observers in a quantum measurement
    • Advanced Physics Homework Help
Replies
3
Views
1K
Comp Sci An Introduction to Deep Learning and Modifying Code
    • Engineering and Comp Sci Homework Help
Replies
7
Views
875
Model for a Qubit system using the Hamiltonion Operator
    • Advanced Physics Homework Help
Replies
6
Views
919
Operators and eigenstates/values
    • Advanced Physics Homework Help
Replies
4
Views
1K
A discrete-time queue Markov Chain problem
    • Calculus and Beyond Homework Help
Replies
12
Views
1K
Constructing an Eigenvector of S with Eigenvalue λ1
    • Advanced Physics Homework Help
Replies
29
Views
2K

Hot Threads

Recent Insights

Back
Top