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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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Question on notation for the inner product of complex vectors

Regarding the wiki: https://en.wikipedia.org/wiki/Sesquilinear_form#Hermitian_form you can see that the wiki states that physics defines the inner product for complex vectors as: $$\langle \, \...
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1answer
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Do Ladder Operators Give All Eigenstates?

The canonical ladder operators for, say, orbital angular momentum are something like $$ \hat L_+ = \hat L_x + i \hat L_y $$ and it can be shown that, if $ \left| \phi \right> $ is an eigenstate ...
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62 views

What does the notation $\langle a|b|c\rangle$ mean? [on hold]

What does the notation $\langle a|b|c\rangle$ mean? I saw this in a Quantum Mechanics book and couldn’t understand it
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1answer
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How Do We Define Integration over Bra and Ket Vectors?

I'm having trouble understanding the completeness condition for bra and ket vectors in Hilbert space, especially in the continuous case. The discrete case makes a fair amount of sense; given any ...
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1answer
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Integrating of von Neumann equation for density matrix

Suppose we are given the Hamiltonian $$H=f \frac{\text{Tr}\sigma_x \rho}{\text{Tr}\rho}\sigma_x,$$ where $\rho$ is the density matrix, and $\sigma_x$ is the Pauli matrix $$ \sigma_x= \begin{...
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1answer
49 views

Effects of measurement on particle energy

According to quantum mechanics, once you measure a particle's energy, its wave function collapse into some state, an eigenfunction with some eigenvalue (which is the particle energy). But if a ...
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Intuitive understanding of partial trace

Suppose we have some quantum system with sub-systems A and B. It could be, for example, two qubits or groups of qubits. Is it fair to say that tracing out the sub-system A is always akin discarding ...
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2answers
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What does $e^{i\alpha}$ stand for in the general expression for a qubit gate $e^{i\alpha}R_n(\theta)$?

All qubit gates can be written in the form of: $$U = \exp(i\alpha)R_n(\theta).$$ I know $R_n(\theta)$ is a rotation $\theta$ about an arbitrary axis, but what does $\exp(i\alpha)$ stand for? From my ...
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Doubt from Momentum operator in the position basis [duplicate]

J.J Sakurai shows in the section of ' Momentum operator in the position basis' as $$P \lvert\alpha\rangle = \int dx^{'}\lvert\ x{'}\rangle\Bigl(-i{h\over 2\pi} {\partial\over\partial x} \langle\ x{...
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1answer
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Operators acting on a single subsystem within a combined system's state

I was reading over combined systems and operators acting on a single system within the combined system, and am confused by the math. For example, we have individual qubit states for subsystems $A$ ...
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Generalisation of finite to infinite vector space [closed]

To generalise finite vector space into infinite vector space..this book(principle of quantum mechanics) says we redifine inner product of 〈f|g〉as [ limit n tends to infinity Σfₙ(xᵢ).gₙ(xᵢ)Δ ]where ...
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Questions about BRST symmetry [closed]

For a course about the standard model, I am writing a paper on BRST symmetry. For this I am mainly following the material developed in chapter 16.4 of Peskin and Schroeder. I am mostly done, however ...
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Can someone please explain the meaning of the circled paragraph?

why does the off diognal elements of the matrix mediate with the coupling differential equation?
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2answers
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Why can $|\Psi (t=0)\rangle $ be written as a coherent superposition of some eigenkets?

Why can $|\Psi (t=0)\rangle $ be written as a coherent superposition of some eigenkets? One of the approaches to solve time dependent Schrodinger equation $i\hbar \frac{\partial |\Psi(t)\rangle}{\...
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1answer
60 views

What are the eigenstates of $X^N$ operator?

The operator $X$ is the position operator with it's conjugate being the momentum operator: $$[X,P]=i$$ ($\hbar=1$). Eigenstates of the position operator is known as quadrature/position states: $$X|x\...
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1answer
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Partitioning Fermionic Systems

When dealing with separate non-interacting systems in quantum mechanics (i.e. with distinct Hamiltonians acting on different Hilbert spaces like $\hat{H}=\hat{H}_1 + \hat{H}_2$), one can often write ...
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Applying the adjoint of an operator

Consider the following inner product: $$ \langle x | Z\rho Z^\dagger | y\rangle$$ Here $Z$ is an operator and $Z^\dagger$ is it's conjugate. $\rho$ is a density matrix. Does this equal to the ...
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1answer
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Why do we need creation and annihilation operators in QFT?

2. Why do we need creation and annihilation operators? Main point is that a particle can be created by creation operator and destroyed by ...
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1answer
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Question about understanding quantum fields [closed]

1. How do we interpret colisions in QFT formalism? How did we know, when developing the theory, that we are getting the fields which describe creation of particles? How does one excite the field to ...
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2answers
104 views

Is there a difference between a Hermitian operator and an observable?

My poorly written lecture notes say that any Hermitian operator does have a complete set of orthogonal eigenstates with real corresponding eigenvalues and is therefore an observable. In the article ...
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1answer
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Is the scalar product also a wave function?

I am wondering about the meaning of the scalar product and its relation with the wave function. In the Hilbert space, the scalar product is defined as $$\langle \phi \rvert \psi \rangle = \int \phi^*\...
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1answer
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Path integral formulation of amplitude from initial to final state

In path integral formulation we say that we are summing over all possible ways for the system get from initial to final state. Now if we just write the amplitude and then insert complete set of states,...
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Orthogonalizing a Gaussian Basis

Given a discrete Gaussian basis $$G = \{\lvert n\rangle, n \in \mathbb{Z}\},$$ where $$\langle x\rvert n \rangle = \exp\left(\dfrac{-(x-nL)^2}{2}\right),$$ with $L$ fixed. Does there exist a set of ...
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3answers
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What does $X|n\rangle \propto |n+c\rangle$ mean?

$\renewcommand{\ket}[1]{\left \lvert #1 \right\rangle}$ I'm transcribing below (but see edit history for a scan) a calculation from pg 17 of this article on Lie groups and Lie algebras $$ [N,X]=cX....
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1answer
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Orthogonality of quasibound states

Suppose you have a simple 1D problem with a potential which is such to allow for bound, quasibound, and free states. Are the quasibound states orthogonal to the bound states, or is there some slight ...
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1answer
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Expectation value of derivative of operator

I was given the following question: Let $A(\lambda)$ be a Hermitian operator, which is dependent on some real parameter $\lambda$. Let us denote the eigenvalues and corresponding eigenstates of $A$ ...
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1answer
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Question about field quantization

I'm reading a paper by Rubin, Klyshko, Shih and Sergienko titled "Theory of two-photon entanglement in type_II optical parametric down-conversion", 1994 (link to the paper). I'm stumped by equation ...
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0answers
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Why don't we normalize the wavefunction in time? [duplicate]

I am aware that because a particle whose wavefunction we are dealing with must be found somewhere, we normalize the probability density in position. Why do we not normalize the wavefunction in time? ...
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1answer
29 views

The differentiation of a functional by a ket

I saw something very strange when I was studyng about the variational method. In the text, to minimize the functional $$E[\psi] = \frac{\langle \psi |\hat{H}|\psi \rangle}{\langle \psi |\psi \rangle},$...
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3answers
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Does the set of the degenerated eigenfunctions of hamiltonian forms a subspace?

I have read in a book that the set $\{ \Psi_{n}^{(\nu)} \in \mathcal{H} | \ \ \hat{H}\Psi_{n}^{(\nu)} = E_{n}\Psi_{n}^{(\nu)} \}$ (that is, the set of all eigenfunctions of the hamiltonian with the ...
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1answer
60 views

Why are coherent states not linearly independent?

From the completeness relation one can see that, $$|\psi \rangle = \int \frac{d^2 \alpha}\pi \langle \alpha | \psi \rangle |\alpha\rangle.$$ And if $|\psi\rangle = |\beta \rangle$ (which is another ...
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2answers
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Wigner-Eckart theorem and vectors

Let's consider a system in state $^3$D$_1$: $$\vec{L}^2=L(L+1)=6 $$ $$\vec{S}^2=S(S+1)=2$$ $$\vec{J}^2=J(J+1)=2$$ According to Wigner-Eckart theorem, if this is an irreducible representation, all ...
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3answers
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What are the eigenvector's of the $\hat a^2$ operator?

Since $\hat a^2$ and $\hat a$ commute, then one of the eigenvectors of $\hat a^2$ will be, the coherent state $|\alpha\rangle$. Are there others states as well?
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1answer
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Derivative of tensor product of quantum states

Recently I asked a question over at the math stack exchange: https://math.stackexchange.com/q/3210375/. However I figured I'd ask here too, seeing as the question originated in a physics course I'm ...
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1answer
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Why can the partial trace be written as $\text{Tr}_B(\rho)= \sum_k (1 \otimes \langle k|) \rho (1 \otimes |k \rangle)$?

I don't really understand a notation that I stumbled upon regarding a partial trace. According to the definition I have, partial trace is $$\rho_A=\text{Tr}_B(\rho_{AB}):= \sum_k (1_A \otimes \...
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Can one add a discrete set of functions to complete the bound states of the hydrogen atom?

Though being an infinite orthonormal set of functions, the bound states $\Psi_{nlm}$ of the hydrogen atom do not form a basis of the Hilbert space $L^2(\mathbb{R}^3)$ due to the continuous spectrum, i....
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2answers
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Transformation connecting two representations - Quantum mechanics [duplicate]

I am working on Dirac's paper The Lagrangian in Quantum Mechanics. He looks for the analogy between a classical transformation between two sets of coordinates and momenta $p_r$, $q_r$ and $P_r$, $Q_r$ ...
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1answer
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Derivating operator acting on ket

I'm deducing a formula, and I used the "product rule" $\frac{\partial}{\partial t}(A|\phi>)=(\frac{\partial A}{\partial t})|\phi>+A\frac{\partial}{\partial t}|\phi>$. I'm actually getting the ...
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2answers
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Some quantum-mechanical questions [closed]

I have recently started studying quantum mechanics, and here are some things that are really confusing me. Particle in a box: Supposedly, the square of the magnitude of the normalized wave function ...
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1answer
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Reverse Clesbch Gordan coefficients

I'm quite unsatisfied with this answer, so I'm hoping to get an adequate one. I'm trying to understand how to compute the reverse CB coefficents. I'll provide the simplest example. If I have a $l =...
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1answer
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Constraints on higher-dimensional Bloch vectors

I'm interested in the constraints on the $(4^n-1)$-dimensional generalized Bloch vector (the Bloch vector for $n$ qubits). To the best of my knowledge, these are not analytically characterized for ...
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1answer
55 views

Which properties does tensorial product have with respect to scalar product?

Are the associative and distributive properties preserved? In Cohen-Tannoudji's Mécanique quantique vol. I, the scalar product in the $\epsilon=\epsilon_1 \otimes\epsilon_2$ space is defined as ...
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2answers
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Why can an inner product of an eigenvector also be used as an eigenvector?

In quote box below, there is an inner product of an angular momentum eigenvector. Why can you use this inner product as a new eigenvector for the next part of the work? And why do they "of course" ...
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2answers
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Precise definition of the Hilbert space in QM?

In QM books (at least those I have read) the definition of the Hilbert space used is somewhat blurred (the "space of square integrable functions" is not enough to define it precisely : which kind of ...
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1answer
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Proof for $\langle i[A,B]\rangle$ [closed]

I have to prove the following equation: $$ \langle i[A,B]\rangle = 2\mathfrak{Im}\left[\int dV(\overline{B\psi)}(A\psi)\right]\,,$$ where A,B are hermitian operators. Here is my calculation, but I don'...
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How to understand notation in “Introduction to Quantum Mechanics (3rd Edition)” by David Griffiths, Chapter 3.6.2?

In the 3rd edition, on page 118, the projection operator is introduced as $$\hat{P}=|\alpha\rangle\langle\alpha|.$$ Then Griffiths says that when $\hat{P}$ acts on another vector, it looks like ...
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Source for mathematical methods [duplicate]

I am just curious about any question and example sources for linear vector spaces, bra-ket notation, operators, commutators and hilbert spaces.
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3answers
36 views

Symmetry of the spin function and T0 and S states

$|T_0\rangle = \frac{1}{\sqrt{2}}(|\uparrow \downarrow\rangle + | \downarrow\uparrow\rangle )$ is a triplet state, whose spin function has to be symmetric. $|S \rangle = \frac{1}{\sqrt{2}}(|\...
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1answer
74 views

Decomposition of the maximally entangled states

We know that the set of symmetric bipartite pure states is spanned by $S=\{|\phi\rangle^{\otimes 2},|\phi\rangle \in \mathbb{C}^d\}$. I want to know if the maximally entangled state $|\psi\rangle = \...
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1answer
52 views

Matrix representation of spin-1/2 operators in Sakurai

Hello and thanks for reading. I'm an undergrad working through the first chapter of Sakurai's text and was going through the principles of the spin-1/2 system. The author demonstrates closure and ...