# What is Unitary transformation: Definition and 19 Discussions

In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.

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1. ### A Similarity transformation changing the determinant to 1

In Michael Tinkham's book, Group theory and Quantum Physics, he derives a theorem that any matrix representation can be converted to an equivalent transformation which is unitary. i.e ##A## is converted to ## B = S^-1 A S ## such that B is unitary. My question is how is it possible to find such...
2. ### Unitary vector commuting with Hamiltonian and effect on system

Hi, I'm not sure to understand what ##| \phi_n \rangle = \sum_i \alpha_i |\psi_n^i## means exactly or how we get it. From the statement, I understand that ##[U,H] = 0## and ##H|\psi_n \rangle = E_n|\psi_n \rangle## Also, a linear combination of all states is also an solution. If U commutes...
3. ### Unitary transformation using Python

I would like to ask about unitary transformation. UA(IV) UB*UA(IV) UAT(UB*UA(IV))=UB(IV) UB(IV)*(X) IVT(UB(IV)*(X))=UB(X) UBT*UB(X)=X From the information above, UAT,IVT and UBT are the transpose of the complex conjugate. The aim of this code is to get the value of X in the step 4. This is...
4. ### I How can I find the unitary matrix for diagonalizing a Hamiltonian numerically?

Hi! I'm trying to understand how to diagonalize a Hamiltonian numerically. Basically I have a problem with a Hamiltonian such as H = \frac{1}{2}c^{\dagger}\textbf{H}c where c = (c_1,c_2,...c_N)^T The dimensions of the total Hamiltonian are 2N, because each c_i is a 2 spinor. I need to...
5. ### I Change of Basis and Unitary Transformations

Say, we have two orthonormal basis sets ##\{v_i\}## and ##\{w_i\}## for a vector space A. Now, the first (old) basis, in terms of the second(new) basis, is given by, say, $$v_i=\Sigma_jS_{ij}w_j,~~~~\text{for all i.}$$ How do I explicitly (in some basis) write the relation, ##Uv_i=w_i##, for...
6. ### Finding a unitary transformation between two quantum states.

I have to find a unitary transformation that takes me from one quantum state to another (or if there is such a transformation), given the two quantum states in matrix form. The matrices are huge (smallest is 16x16) , so doing it on paper is not an option. Does anyone know how I can do this in...
7. ### Finite and infinite unitary transformation

hi, i know unitary transformation - but could not get where do we need finite and infinite unitary transformation ? please help me in this regard. thanks
8. ### Unitary transformation of pure states to other pure states

Is it true that there always exist a unitary matrix that can take a state vector of an arbitrary pure state to another arbitrary pure state ? (of course assuming same hilbert space). If true, how do we prove it ? it look like it is true via geometrical arguments but i have not been able to...
9. ### Simple question concerning unitary transformation

Is the transformation of an operator under INFINITESIMAL unitary transformation, U^-1AU or UAU^-1?? I saw that two books defined it differently?
10. ### Diagonalize matrix by unitary transformation

In an exercise I am asked to find the eigenvalues of a matrix A by demanding that a unitary matrix (see the attached file) diagonalizes it. I know I could just solve the eigenvalue equation but I think I am supposed to do it this rather tedious way. Now I have introduced an arbitrary unitary...
11. ### MHB Jordan Normal Form of Unitary Transformation

Hi everyone, :) Recently I encountered the following problem. Hope you can confirm whether my method is correct. My answer seems so trivial and I have doubts whether it is correct. Problem: Find the Jordan normal form of a unitary linear transformation. My Solution: Now if we take the...
12. ### Two component field theory and unitary transformation

I have a theory described by a 2-component field \psi_i (i'm working with BCS in Nambu-Gorkov representation, but any other field theory would be ok, that's why I'm posting in this subforum), and the lagrangian it's defined in the following way: \mathscr{L}=\psi^\dagger \Gamma \psi where...
13. ### Unitary Transformation: Proving ¯UU = 1 in Dirac's Text

In Dirac’s text the equation ¯UUα=α¯UU is well proven . Next it is said that since ¯UU commutes with all linear operators so it must be a number . Further since ¯UU and its complex conjugate are same so ¯UU is a real number . Also Dirac mentions that for any ket |P> , <P|¯UU |P> is positive...
14. ### How can I apply a unitary transformation to rotate a 3D complex wave function?

Hello, I have a 3D complex wave function and I want to apply a unitary transformation to rotate it with respect to arbitrary axis. Anybody have any ideas how I can do that? Sasha
15. ### Physical interpretation of unitary transformation

what actually happens physically ...when we make transpose of a matrix...and in unitary transformation we transpose the matrix and take the conjugate...physically what type of change happens in it.
16. ### Unitary Transformation Problem

Homework Statement Hi guys, I'm working on a question in Tom M Apostol's second calculus book, on page 141, number 5. It is: Prove that if T (an operator on a vector space V) is linear and norm-preserving, then T is unitary. Homework Equations Okay, a transformation T is linear if it...
17. ### Finding unitary transformation

Homework Statement Find a unitary transformation that diagonalizes the matrix: 1 1 1 -3 1 1 1 -3 1 1 1 -3 -3 -3 -3 -9 Homework Equations The Attempt at a Solution So before I even start with finding the eigenvalues for this, I know there has to be...
18. ### Similarity vs. Unitary Transformations

Can it be said that similarity transformation is a transformation in real space while unitary transformation is a transformation in complex space?
19. ### About symmetry operation and a corresponding unitary transformation

For each symmetry operation R acting on a physical system,there is a corresponding unitary transformation U(R). But what is the principle for such relation? an example is that : for a continuous symmetry ,we can choose R infinitesimally close to the identity ,R=I+eT ,and the U is close to I...