QM Linear Algebra: Compute Matrix Exponential & Verify Unitarity

In summary: In short, the author does not know what the given matrix is, and does not know what unitary means. They suggest looking for the diagonalized form of the matrix and using that to find the U matrix. Once they have the U matrix, they should be able to evaluate the answer to the given equation.
  • #1
cscott
782
1

Homework Statement



I'm given the Taylor expansion of [itex]e^{i\Theta M}[/itex]
Then the question says "Compute the exponential for the given matrix M and verify that the resulting matrix is unitary.

The Attempt at a Solution



I really just don't know what they're asking...
 
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  • #2
What's the given matrix? And do you know what unitary means?
 
  • #3
Given matrix [[3/4, 1/4][1/4, 3/4]] and yeah I know what unitary means.
 
  • #5
Yes it is.

I just really don't know what they're asking me to do.
 
  • #6
So if you have the expression for exponentiating a matrix, and the matrix to exponentiate, it seems to me like you can do it. What's the problem?
 
  • #7
They give me the Taylor expansion for that exponential (I was just too lazy to latex it)... wouldn't it be stupid to evaluate a random amount of terms?

[tex]e^{i\theta M} = \Sigma\frac{\left(i\theta\right)^n}{n!}M^n[/tex] from n=0 to infinity
 
  • #8
I suppose having the diagonalized version of the matrix would help for raising it to a power.

Is it correct that the diagonalized version would be [[2, 0][0, 1]]?
 
  • #9
If you put M into the form UDU* where U is a unitary matrix, D is diagonal, and U* is the inverse (or transpose) of U, does that help?

And no, that diagonalised form is not quite right.
 
Last edited:
  • #10
Mmm... how about D= [[1, 0][0, 1/2]] ?

Is there a maple command for doing this?
 
  • #11
I don't know about Maple. But it's pretty easy to do by hand -- it's not like a 2x2 matrix is that big...
 
  • #12
genneth said:
I don't know about Maple. But it's pretty easy to do by hand -- it's not like a 2x2 matrix is that big...

Yeah I know but I just wanted a quick way to verify it.
 
  • #13
cscott said:
Yeah I know but I just wanted a quick way to verify it.

In the diagonal form, the elements are the eigenvalues. To find the U matrix I mentioned above, you'll need the eigenvectors anyway. So just act M on the eigenvectors and you should end up with the respective eigenvalues. That should be check enough, and quick.
 
  • #14
I'm pretty sure the answer is D= [[1, 0][0, 1/2]] but if I don't know how high n should go how can I get a matrix out of the Taylor expansion they gave me?
 
  • #15
n is for all n: 0 to infinity. D^n for any diagonal matrix should be easy to evaluate. U*U = I for unitary matrices. And remember that matrix multiplication is distributive: AB+AC=A(B+C). Does that help?
 

FAQ: QM Linear Algebra: Compute Matrix Exponential & Verify Unitarity

What is the matrix exponential and why is it important in quantum mechanics?

The matrix exponential is a mathematical function that takes a square matrix as its input and outputs a new square matrix. In quantum mechanics, the matrix exponential is used to describe the time evolution of quantum systems. It is important because it allows us to calculate the state of a quantum system at any given time, which is crucial for understanding and predicting the behavior of quantum particles.

How do you compute the matrix exponential in linear algebra?

The matrix exponential can be computed by using the Taylor series expansion, which involves summing an infinite series of terms. Alternatively, there are also efficient algorithms and software packages available for computing the matrix exponential, such as the Padé approximation method.

What does it mean for a matrix to be unitary in quantum mechanics?

In quantum mechanics, a unitary matrix is a square matrix that preserves the inner product of two vectors. This means that the matrix does not change the length or angle between two vectors, which is important for maintaining the probabilistic nature of quantum systems. In other words, a unitary matrix represents a reversible transformation of a quantum state.

How can you verify if a matrix is unitary?

To verify if a matrix is unitary, you can check if its inverse is equal to its conjugate transpose. In other words, if the matrix A is unitary, then A†-1 = A*, where A* is the conjugate transpose of A. Additionally, you can also check if the columns (or rows) of the matrix are orthonormal, meaning that they are orthogonal to each other and have a length of 1.

Can you give an example of a unitary matrix in quantum mechanics?

One example of a unitary matrix in quantum mechanics is the Hadamard gate, which is a 2x2 matrix used in quantum computing. It represents a basic quantum operation that can be used to create superposition states and perform quantum measurements. The Hadamard gate is unitary because its inverse is equal to its conjugate transpose.

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