Problem 3. Find the unitary operator hat{U} which diagonalizes one of the Pauli matrices hat{sigma}_x = begin{pmatrix}0 & 1 1 & 0 end{pmatrix} Obtain the eigenvalues and eigenvectors of hat{sigma}_x

Lewis Harvey 2021-01-04 Answered
Problem 3. Find the unitary operator U^ which diagonalizes one of the Pauli matrices
σ^x=(0110)
Obtain the eigenvalues and eigenvectors of σ^x
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

bahaistag
Answered 2021-01-05 Author has 101 answers

Step 1
Given
Pauli matrices
σ^x=[0110]
According to question
U^σ^x=[λ100λ2]
Where U^ is the unitary operator which diagonalizes the the Pauli matrices.
Step 2
Now Eigenvalues of Pauli matrix.
|σ^xλI|=[λ11λ]=0
[λ11λ]=0λ21=0
λ=1,1
Now Eigenvector,
λ=1
[1111][xy]=0
x=y
Hence Eigenvector for λ=1 is
X=[11]
Eigenvector for
λ=1
[1111][xy]=0
x=y
Hence Eigenvector for λ=1 is
Y=[11]
Let for unitary operator,
U^=[abcd] then
[abcd][0110]=[1001]
[abcd]=[1001][0110]1
[abcd]=[0110]
U^=[0110]

Not exactly what you’re looking for?
Ask My Question
Jeffrey Jordon
Answered 2022-01-24 Author has 2047 answers

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more