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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Time evolution of the standard deviation of an operator

How would I find the time evolution of the standard deviation of an operator? For example, how might I find the time evolution $\sigma_x (t)$ of the standard deviation $\sigma_x = \sqrt{ \langle \hat{...
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What are we measuring in a quantum field when we square the wavefunction?

Suppose we are doing a measurement in a particular quantum field, i.e electron field. Are we looking for the probability of the electron to show up at that spot we are measuring or are we measuring ...
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Superposition of two coherent states in QFT

Let's suppose we have a scalar bosonic field in a coherent state configuration (for a single mode m). In Fock space, this would be represented as a coherent superposition of Fock states of all ...
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Need some help with Bra-Ket notation (specifically orthogonality in bra-ket notation)

I'm reading notes from a friend of mine taking a quantum mechanics class, and I see something I don't quite get. $$\left<x_i|x_j\right> = \delta_{ij}.$$ The notes say this implies orthogonality. ...
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Individual particle states in Fock space

I am currently learning QFT, and after watching the wonderful lectures by Leonard Susskind (https://theoreticalminimum.com/courses/advanced-quantum-mechanics/2013/fall), I am still struggling to see ...
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Infinite correlation functions in free field theory

In a free scalar field theory, Wick's theorem guarantees that $\langle \hat\phi(x)\rangle = 0$ and $\langle \hat\phi(x)^2\rangle = \infty$. Given that $\hat \phi(x)$ creates a particle at $x$, these ...
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A Projection Operator generated by the Eigenspaces of an Observable?

What does it mean for a projection operator on to a subspace of Hilbert space for a system $S$, $H_{S}$, to be generated by the eigenspaces of an observable $A$ that correspond to a certain set $\...
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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 ...
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Confusion on the mathematical aspect of the Translation operator

I know that the Translation operator shifts particles/fields in a direction and can be written as $$\hat{T}(x) = e^{-\frac{i}{\hbar}\hat{p}x}.$$ What confuses me is that when we rewrite $\hat{p} = \...
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Physical significance of this operator in quantum mechanics

I have stumbled across this question and cannot seem to find an answer to it. Consider an operator $\textbf{A}$ with eigenkets $|{a_{i}\rangle}$ and distinct eigenvalues $a_{i}$ . One can check that ...
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Are density states more fundamental than wavefunctions? [duplicate]

Some interpretations, like the many-worlds interpretation, treat the wavefunction (modulo an overall phase factor) as objective and fundamental. But consider the following example for a qubit: a ...
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How can we write creation and annihilation operators in first quantized notation and second quantized notation?

We have learned creation operator $\hat{a}^\dagger_i$ adds a particle in $i^{th}$ state, and annihilation operator $\hat{a}_j$ remove a particle from $j^{th}$ state. They can be interpretated in such ...
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Defining Superposition

I’d hope to keep this simple. If we decide to measure electron spins along the vertical axis using a Stern-Gerlach device we get an overall even split between up and down. Therefore it is said ...
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Wave function as a ket vector in a Hilbert space

There's something I don't understand: I've learned that quantum wave functions can be described as a "ket vector" in an abstract vector space called Hilbert space. The position wave function,...
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Dividing the Hilbert space, two different cases in analogy

In this post I'll write two different cases that are conceptually different but to me they are analogous. I'd like if you give me a unified view of this cases. In the Schrodinger equation for central ...
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The physical meaning of ${\rm Tr}()$ in quantum mechanics [duplicate]

I know what trace is in matrix algebra, but does it have any intuitive physical significance when acted on a quantum mechanical operator? What aspect of an operator $A$ does ${\rm Tr}(A)$ capture? Or ...
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Understanding superposition principle

Does the superposition principle actually tell us about our inability to predict what happens during the course of the experiment? Does it tell that, since an experiment has multiple outcomes ( i.e , ...
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Wick's theorem contributions to nucleon scattering

I am following David Tong's notes on QFT. On page 58, he applies Wick's theorem to $\psi\psi\rightarrow\psi\psi$ scattering for a scalar field with a Yukwara interaction term. The claim is, that by ...
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Calculating $S_z$ operator in $S_y$ basis. What's wrong with my approach?

I want to find the matrix representation of $S_z$ operator in terms of the eigenkets of $S_y$ operator. ( It's Problem 1.24 (b) of Sakurai Second Edition ) One simple way to solve this problem is to ...
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Is the Schrodinger's Cat experiment possible in principle?

There are already lots of questions (and great answers) regarding Schrodinger's Cat (and Wigner's Friend, which is the same concept) on here. For example, this post is a great explanation of how ...
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Is this integral always equal to 1?

This is my Hamiltonian. $\psi_{\alpha}$ is a bosonic field. $$H_{\alpha}=\int \mathrm{d} \mathbf{r} \psi_{\alpha}^{\dagger}(\mathbf{r})\left(-\frac{\nabla^{2}}{2 m}\right) \psi_{\alpha}(\mathbf{r})+\...
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Is this “position” operator $\hat{A}$ defined in the whole Hilbert space $L^2(\mathbb{R})$?

I was reading about bounded operators, and I was wondering that if we assume a "position" operator $A$ that acts on $\psi(x)$ and here $x$ is bounded in the interval $[-1,1]$ for example. ...
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What does it mean for a projection operator to represent a state?

I can understand that an idempotent operator can be represented as a projection operator, such as $|x\rangle\langle x|$. But some authors seem to use projection operators, instead of vectors, to ...
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Feynman diagrams and pair creation/annihilation

I'm trying to make sense of QFT (I am a mathematician). I have absorbed "from the ether" the following physical interpretation of Feynman diagrams in the Hamiltonian picture, which I really ...
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Dimensionality of a Hilbert Space: Countable or Uncountable? [duplicate]

(My question is different than this one and the similar one about the free particle, so hold back on casting a close vote, please). So, I was reading on Wikipedia, and ran into this statement in the ...
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Where does the imaginary unit $i$ come from in representing spin vector along y axis?

I am currently reading Leonard Susskind's - "Quantum Mechanics - The Theoretical Minimum". On Page 38 of the book, the writer described representing spin vectors along the $x$- and $y$-axes ...
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Significance of Diagonalization in Degenerate perturbation Theory

I am studying Degenerate perturbation Theory from Quantum Mechanics by Zettili and i'm trying to understand the significance of diagonalizing the perturbed Hamiltonian. He uses the stark effect on the ...
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When does a Hermitian operator have real matrix elements?

I will use braket-notation, but my question is not specific to quantum mechanics. Instead, I would be interested in a general answer for operators in some Hilbert space. Let $H$ be a Hermitian ...
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In deriving the expression for the position operator in momentum space

Consider this: $$\langle \mathbf{r} | \hat{\mathbf{P}} | \psi \rangle = \displaystyle\int d^3\mathbf{r}'\displaystyle\int d^3\mathbf{r}''\langle \mathbf{r}|\mathbf{r'}\rangle\langle\mathbf{r}'|\hat{\...
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1answer
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Physical interpretation of a Hamiltonian [closed]

In natural basis $| 0 \rangle = \begin{pmatrix} 1 \\0 \end{pmatrix}$, $| 1 \rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, what physical situation/model does the following Hamiltonian represent: $H = ...
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I'm confused about two aspects of quantum mechanics: 1.probabilistic nature of quantum 2.time evolution operator [duplicate]

first >> we know that quantum mechanics works with a probabilistic nature so that we can't say " what will happen? " but " what might happen? " second >> we can ask how ...
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Does the vacuum state exist in reality?

If I understand it correctly, in Quantum Field Theory, a vacuum state is a state with zero particles in each mode. However, if a photon is "created" with a specific momentum, it will be ...

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