Expectation values and trace math

In summary, to get the expectation value of operator \sigma using density matrix \rho in a trace, you can take the matrix product \Lambda\rho and then sum the diagonal entries to compute the trace.
  • #1
cscott
782
1

Homework Statement

How do I get the expectation value of operator [itex]\sigma[/itex] using density matrix [itex]\rho[/itex] in a trace: [itex]Tr\left(\sigma\rho\right)[/itex]

I have [itex]\sigma[/itex] and [itex]\rho[/itex] in matrix form but how do I get a number out of the trace?
 
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  • #2
cscott said:

Homework Statement




How do I get the expectation value of operator [itex]\sigma[/itex] using density matrix [itex]\rho[/itex] in a trace: [itex]Tr\left(\sigma\rho\right)[/itex]

I have [itex]\sigma[/itex] and [itex]\rho[/itex] in matrix form but how do I get a number out of the trace?

I am not sure I follow your question. Do you know what it means to take the trace of a matrix?
 
  • #3
I do if it involves just a bra and ket

i.e. [tex]Tr\left(|a><b|\right) = <b|a>[/tex]

I've been shown [itex]<\Lambda> = Tr\left(\Lambda\rho\right)[/itex]

But I have rho and lambda in matrix form and not as a product of bra's and ket's
 
  • #4
The trace is additive, so

[tex]
\mathrm{Tr}\left( \sum_i | a_i \rangle \langle b_i| \right) = \sum_i \mathrm{Tr}\left( | a_i \rangle \langle b_i| \right)
[/tex]

If you have a matrix, this greatly simplifies. It's just the sum of the diagonal entries.
 
Last edited:
  • #5
Hurkyl said:
The trace is additive, so

[tex]
\mathrm{Tr}\left( \sum_i | a_i \rangle \langle b_i| \right) = \sum_i \mathrm{Tr}\left( | a_i \rangle \langle b_i| \right)
[/tex]

If you have a matrix, this greatly simplifies. It's just the sum of the diagonal entries.

Ahh I remember that now.

So I just take the matrix product [itex]\Lambda\rho[/itex] and then sum the diagonal entries to compute the trace?
 
  • #6
I got the correct answer. Thanks guys.
 

1. What are expectation values in math?

Expectation values in math are a way to measure the average value of a random variable. It is calculated by taking the sum of all possible outcomes of a variable multiplied by their respective probabilities.

2. How are expectation values used in statistics?

In statistics, expectation values are used to calculate the mean of a probability distribution. This allows for a better understanding of the data and can help in making predictions about future outcomes.

3. What is the significance of expectation values in quantum mechanics?

In quantum mechanics, expectation values represent the average measurement result of a physical quantity in a given state. They are used to predict the behavior of quantum systems and are an important tool in understanding the probabilistic nature of quantum mechanics.

4. What is the difference between expectation values and average values?

Expectation values and average values both represent the average of a set of values. However, expectation values take into account the probabilities of each value occurring, while average values simply calculate the sum of all values divided by the number of values.

5. How is the trace used in calculating expectation values?

The trace of a matrix is used in calculating expectation values by taking the sum of the diagonal elements of the matrix. In quantum mechanics, the trace represents the sum of all possible outcomes of a variable, making it a useful tool in calculating expectation values for quantum systems.

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