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Tagged with unitarityhomework-and-exercises
24 questions
1vote
1answer
95views
Writing a given unitary in the same basis as the Hamiltonian (Operator Representation and Confusion)
I have a simple question concerning how to write the representation of operators, such as unitaries, using a specific order for the basis elements. Let me give you an example. Consider a tripartite ...
- 110
2votes
1answer
107views
Effect of applying Hermitian conjugate of inversion operator
I'm glad Victor Galitski's monolith is finally out in English version, but I was confused by the following problem: $$ \text{Find the Hermitian conjugate of the inversion operator }\hat{\Pi}: \hat{\Pi}...
- 137
0votes
1answer
137views
How to make this matrix unitary?
I have the following matrix from quantum mechanics, $X = a_0 + \sigma\cdot\mathbf{a}$, where $\sigma$ are the usual Pauli matrices. I can expand this into a matrix form of $X$, $$X = \begin{bmatrix} ...
- 1,073
2votes
1answer
526views
Finding relation between matrix $S$ and matrix $M$ for wave propagation
we have the same Scattering matrix concept in RF as in quantum physics however, I couldnt derive an expression for the $S$ matrix using the $M$ matrix elements and vice-versa. How can I derive eq 1.13 ...
- 33
0votes
1answer
462views
Prove that the scattering operator is unitary [duplicate]
Let $H_0$ be some initial time-independent hamiltonian, and let $V$ be a scattering potential, such that the hamiltonian of a scattering process is: $$H=H_0+V$$ Define the quantum states $|\psi_i\...
- 152
1vote
4answers
254views
Is $U^\dagger(R)\hat{H}U(R)=\hat{H}$ always true?
Consider a Rotation transformation on momentum state, $$U^\dagger(R)\hat{\mathbf{p}}U(R)=R\hat{\mathbf{p}}$$ Now the question is whether, $$U^\dagger(R)\hat{H}U(R)=\hat{H}\,?$$ Here, $\hat{H}$ is the ...
- 814
0votes
1answer
276views
Proof that $U$ is an unitary operator [closed]
I have a function $f$ mapping a bit onto another bit, i.e. $f : \{0, 1\} \rightarrow \{0, 1\} $. The function f is either constant, so f(0) = f(1) or balanced, so f(0) $\neq$ $f$(1). The quantum gate ...
- 409
4votes
1answer
1kviews
Prove that there exists a $d \times d$ unitary matrix $U$ which cannot be decomposed as a product of fewer than $d-1$ two-level unitary matrices
I'm trying to solve exercise 4.38 from Nielsen and Chuang, which asks to "Prove that there exists a $d \times d$ unitary matrix $U$ which cannot be decomposed as a product of fewer than $d-1$ two-...
0votes
2answers
2kviews
Commutator under unitary transformation
How can I prove that the commutators are invariant under unitary transformations? I'm studying quantum mechanics, so (maybe) my professor is talking about the commutator of hermitian operators.
0votes
4answers
143views
Why is $\exp \left ( \frac{\pi}{2\hbar}(L_x^2 + L_y^2) \right )$ not a unitary operator? [closed]
I should prove that $$\exp \left ( \frac{\pi}{2\hbar^2}(L_x^2 + L_y^2) \right )$$ is not a unitary operator. Where $L$ is the total angular momentum of a 2-particle system ($L = L_A + L_B$ for the ...
- 177
1vote
0answers
326views
Anti-commutative hermitian operators
I have some trouble to prove the next statements: Let $A,B$ two anti-commutative hermitian operators, i.e. $\{A,B\}=AB+BA=0$. Does $A$ and $B$ share any eigenket?. If $U$ is an unitary operator such ...
- 109
-1votes
1answer
88views
Can someone explain why this matrix is unitary? [closed]
I have a matrix $$ \begin{bmatrix} a & b \\ e^{i\theta}b^* & -e^{i\theta}a^* \end{bmatrix} $$ Where $\theta$ is a real number, and $a$ and $b$ are complex number such that $|a|^2 + |...
- 115
2votes
3answers
2kviews
How to find unitary matrices?
I'm having trouble fully wrapping my head around unitary matrices. I'm working on them in relation to quantum mechanics. The question specifically I am working on is: Given the Pauli matrices $\...
- 173
4votes
4answers
436views
Why is the $\varepsilon^2$ term in an infinitesimal transformation equal to zero?
Given the unitary operator $U=1+i\varepsilon F$ (where $\varepsilon$ is an infinitesimal scalar), in order to prove that $F$ is Hermitian: $$\begin{align} UU^{\dagger} &= 1 \\ (1+i\varepsilon F) (...
- 51
0votes
1answer
1kviews
Unitary rotation of spin states
Consider the $2j+1$-dimensional Hilbert space spanned by the spin states $$\left|j,-j\right>,\left|j,-j+1\right>,\ldots,\left|j,j-1\right>,\left|j,j\right>$$ where $\hbar m$ and $\hbar^...
