How Do You Calculate the Probabilities of Measured Values for a Quantum State?

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.
 

FAQ: How Do You Calculate the Probabilities of Measured Values for a Quantum State?

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.

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.

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.

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.

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

What To Project Upon When Doing QM Measurements?
Replies
9
Views
1K
Probability of the outcomes of ##J^2## and ##J_{z}##?
Replies
4
Views
847
How do you find the probabilities for an anharmonic quantum oscillator state?
Replies
1
Views
533
Two-level Quantum System - initial state
Replies
14
Views
2K
Switching observers in a quantum measurement
Replies
3
Views
1K
Probability of finding a particle in the right half of a rectangular potential well
Replies
16
Views
2K
Most Probable energy after infinite square well expands
Replies
2
Views
775
Operators and eigenstates/values
Replies
4
Views
1K
Possible Results and Probabilities of a Measurement of Operator Q
Replies
7
Views
2K
Probabilities of measuring ##\pm \hbar/2## along ##\hat{n}##?
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top