# Questions tagged [hilbert-space]

Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.

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### When do coordinate transformation preserve orthogonality?

I'm wondering about the situation where you have two orthonormal wave functions, $$\langle \varphi( \vec{r_1}) | \psi ( \vec{r_2}) \rangle = 0 .$$ Under what restrictions would a coordinate ...
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### What do the quantum fields represent, mathematically?

I am looking for insight on quantum field theory, and more precisely, I am interested in having a low-detailed idea of what a quantum field theory is about; moreover, I should say hat I am a ...
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### Determine whether the wave function is acceptable cos X over ( 0. infinity) [closed]

Determine whether the wave function is acceptable if conditions are a) cos x over ( 0, infinity) b) tan theta over ( 0, 2 theta )
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### Dirac's Principles of Quantum Mechanics, Electron Spin

In Section 37 (The spin of the electron), Dirac writes, "The eigenvalues of $m_z$ are $\hbar/2$ and $-\hbar/2$, so the eigenvalues of $\sigma_z$ are 1 and -1, and $\sigma_z^2$ has just the one ...
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### Interpretation of Hilbert space in the Wightman Axioms for QFT

My confusion is about the different Hilbert spaces we meet in QFT. In a first introduction to QFT, the Hilbert space is often taken to consist of wavefunctionals on classical fields on $\mathbb{R}^3$. ...
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### Representing Quantum Gates in Tensor Product Space

I want to write the matrix form of a single or two qubit gate in the tensor product vector space of a many qubit system. Ill outline a simple example: Both qubits, $q_0$ and $q_1$ start in the ground ...
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### Wavefunction and a central potential $V(r)$ that is singular at origin

I read the following line from Weinberg's Lectures in Quantum Mechanics (pg 34): As long as $V(r)$ is not extremely singular at $r=0$, the wave function $\psi$ must be a smooth function of the ...
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### What is the physical intuition for Bloch Sphere? [duplicate]

I am very confused about how to think about the Bloch Sphere. How can we relate the concept of expectation value to the Bloch sphere? If my state lies in let's say $yz$ plane how can we say that ...
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### Hilbert space of two particles of spins $j_1$ and $j_2$, noninteracting versus interacting

For a system of two noninteracting particles of spins $j_1$ and $j_2$, the joint Hilbert space $\mathcal{V}$ is the tensor product of the individual Hilbert spaces $\mathcal{V}_1$ and $\mathcal{V}_2$. ...
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### Can local projections on different parties give the same reduced state?

Suppose I have a bipartite pure state $\vert\psi\rangle_{AB}$. By the Schmidt decomposition, we know that the reduced states $\rho_A$ and $\rho_B$ have the same eigenvalues. I am now interested in ...
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### Is the vacuum state in QFT time independent?

Is the vacuum state in QFT time independent? The ground state of the quantum harmonic oscillator is not...
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### How to know spin out of wave function?

I do not clearly understand some concepts, so maybe someone will clarify this for me. Imagine we have a random wavefunction for an electron, it could be anything. How can I with known wave function ...
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### Why are time derivatives of states in QFT equal to zero?

In equation 6-38 on page 176 of the book "Student Friendly QFT" by Robert D. Klauber it is said that the partial derivative w.r.t. time of a multi-particle state is equal to zero since we ...
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### Understanding bases in quantum mechanics

For $l = 1$ the angular momentum operator $L_z$ has the eigenvalues $\hbar,0,-\hbar$ and the eigenstates are then $|1,1\rangle, |1,0\rangle, |1,-1\rangle$. Now, we can calculate the matrix elements of ...
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### How to tighten the Fuchs-Van de Graaff inequalities for pure and mixed states?

Defining the trace-distance as $D(\rho , \sigma) = \dfrac12 tr|\rho - \sigma|$ and the Fidelity between two quantum states as $F(\rho , \sigma) = tr\sqrt{\sqrt\rho\sigma\sqrt\rho}$ I need to show the ...
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### Units of observables in quantum mechanics

Observables in quantum mechanics are described by Hermitian operators $\hat A: V \to V$, where $V$ is the Hilbert space of states. Examples include the $x$-coordinate operator $\hat x$, the $x$-...
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### Proving 3D Hamiltonian operator is Hermitian

A Hermitian operator $H$ is defined as $$\int f^*(Hg) d^3\vec{r} =\int (Hf)^*gd^3\vec{r}$$ where $f$, $g$ are 3D square integrable functions and the integrals are taken over all coordinates. I am ...
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### Equivalence of Hermitian operator and Hermitian matrix in Quantum Mechanics

I learned that a Hermitian matrix $A$ is defined as a matrix that satisfies $$A^\dagger=(A^*)^\intercal=A,$$ i.e. its Hermitian conjugate $A^\dagger$ is the same as the original matrix $A$. I also ...
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### If superposition is just uncertainty, then how come quantum computers work? [closed]

If superposition is just uncertainty due to a particle changing on observation and not literally 2 things at once, how come quantum computers work while having qubits that are literally 1 and 0 at the ...
42 views