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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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If the dimension of a space is prime, are quantum states in it guaranteed to be entangled?

A rather obvious question perhaps but if I have a Hilbert space of dimension $d$ and $d$ is prime, I cannot possibly write my state as $$\rho = \sum_i p_i\rho_a\otimes\rho_b$$ simply because the ...
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1answer
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Spin and many particle systems

I learned about many particle systems and second quantization recently. The Fock space of distinct particles with single particle Hilbert space $\mathcal H$ was defined to be $\bigoplus_{N=0}^\infty \...
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Superposition of 2 qubit input with a classical $N$-bit basis state

In a problem I am trying to solve, I'm given 2 qubits in the zero state, i.e. |00>, and a classical bit string describing an $N$-bit basis state, |$\psi$>. I need to make a quantum circuit that ...
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2answers
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What are the energy eigenvalues of a particle subject to the potential $V(x)=mg|x|$?

I am considering a particle within a potential given by $$V(x)=mg|x|$$ and am attempting to find the energy eigenvalues of the system. Taking $V(x)$ to be defined piecewise, I've solved the ...
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1answer
56 views

Operators in Dirac notation and matrix representation (intuition)

So, I'm taking a QM 1 course, and we have reached a point where we used Dirac notation to solve two-level systems more efficiently, but our professor never really bothered to explain it further (he ...
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How can we prove the tensor-product basis $\{|j_1,m_1⟩|j_2,m_2⟩\}$ is linearly independent? [on hold]

My reference is Walter Johnson's Book Lectures on Atomic Physics (2006). Given our coupled $J$'s in the equation bellow: $$m|j,m\rangle = \sum_{m1,m2}(m1 + m2)C(j_1,j_2,j;m_1,m_2,m)|j_1,m_1\rangle|...
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1answer
59 views

Where does ket vector live in rigged Hilbert space?

Let's say rigged Hilbert space $(\mathcal{S},\mathcal{H},\mathcal{S}^{*})$ in Gelfand triple. Where would ket vector live in? Would it be $\mathcal{S}$? That is what I thought, but https://arxiv.org/...
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1answer
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Intuitive understanding of a wave function

Looks like wave function is an abstract mathematical object. I was trying to see if there is a simple way to visualize this. Can someone please help with that? I was thinking may be we can think that ...
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2answers
93 views

Calculating expectation value of the Hamiltonian squared

So the main idea of the problem was to find the error in the argument, which I think I have a good grasp of. Basically, the Hamiltonian of the wavefunction is a constant non-zero value inside the box ...
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1answer
48 views

Ladder operators vs creation/annihilation operators

I am trying to figure out the difference between the ladder operators (for harmonic oscillator) $a^\dagger$, $a$ and the creating/annihilation operators $c^\dagger$, $c$. Are they the same? I have ...
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1answer
42 views

Difference, in terms of completeness, between the Dirac well and barrier

I was in my undergraduate QM lecture and we just finished with the Dirac barrier. My question is as follows: We know that the Dirac well’s complete set of solutions requires one bound state and an ...
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Solving a 2x2 Perturbed Hamiltonian Exactly

Problem Consider Hamiltonian $H = H_0 + \lambda H'$ with $$ H_0 = \Bigg(\begin{matrix}E_+ & 0 \\ 0 & E_-\end{matrix}\Bigg) $$ $$ H' = \vec{n}\cdot\vec{\sigma} $$ for 3D Cartesian vector $\...
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1answer
67 views

Eigenvalues and functions in quantum mechanics [on hold]

How do I determine if $\psi(x)$ is a eigenfunction of some operator and find the corresponding eigenvalues, where $\psi(x)$ is the wave function of free particle (potential = zero).
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What is quantum superposition exactly? [closed]

In quantum superposition, what does it mean when 1 object can be in 2 states at the same time.
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124 views

Are eigenstates of the position operator continuous?

The way I've understood it is that eigenfunction of an operator are the different states which the actual wavefunction can take when the property/observable corresponding to the given operator is ...
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What does a basis rotation correspond to physically for linear position-momemtum?

For polarization and angular momentum, rotating the basis corresponds to a very straightforward physical transformation, namely, the physical rotation of an experimental apparatus about an axis in ...
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Why do Physicists call Hilbert spaces Hilbert spaces? [closed]

Personally, I find this usage confusing. Physically speaking a number by itself is meaningless. It's mathematicians that use numbers by themselves. The number give by itself is useful to a ...
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2answers
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Time evolution of stationary states [on hold]

Let's say we have a state $ \phi=\sum_i c_i \phi_i $ where the $ \phi_i $ denote energy eigen vectors with non degenerate eigen values. Now if a measurement of the energy is done this state ...
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2d CFT and first intersection of vanishing curves

In the search for 2d Unitary CFT's, we use the Kac determinant to find the null curves, and remove regions in parameter space(c and h space) where the determinant is negative. Now, for $h>0$ and $ ...
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1answer
60 views

Does a symmetry operator $U$ and its generator $Q$ acting on a vacuum $|0\rangle$ both represent new degenerate vacuum?

After the spontaneous breakdown of a symmetry characterized by $$\hat{U}=e^{i\hat{Q}\theta},$$ the commutation relation $[H,Q]=0$ continues to hold. Let us consider two states after this symmetry ...
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3answers
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Scalar Product and Adjoint Operator in CFT

$\newcommand{ip}[2]{\left< #1, #2\right>} \newcommand{\d}{\, \mathrm d \lambda^n}$For a Hilbert space $(H,\ip \cdot \cdot)$ and an operator$^1$ $A$ the adjoint of $A$ is defined via the ...
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1answer
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Ladder operators and energy levels [closed]

I am studying how to get the normalization factor algebraically in the exited states of the harmonic oscillator using the beautiful ladder operators technique. I am stuck at the point where is stated ...
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1answer
60 views

What does it mean for 2 observables to be compatible?

If I have 2 observable operators $A$ and $B$, if $A$ and $B$ commute: $[A, B] = 0$, then they must necessarily form a complete set of commuting observables (CSCO). Essentially, if 2 observables are ...
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How can one derive that wave function is a Gaussian? [closed]

I've just been looking over Shankar's book on QM and I noticed that he introduces the wave function - the state vector in a position basis - as a Gaussian without any real motivation, whereas he does ...
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1answer
113 views

Path integral in interacting quantum field theory

From my understanding we do not yet know how to make much out of interacting QFT other than scattering amplitude at asymptotic infinity. (Correct me if I misunderstand.) But path integral, in ...
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1answer
144 views

Does Haags Theorem forbid Time-Evolution?

I didn't quite grasp the essence of Haags Theorem in the the way it is presented (for example on wikipedia), but the issue seems to be that if one wants to represent infinitely degrees of freedom ...
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1answer
51 views

Polchinski's No-Ghost Proof

I'm struggling to understand an aspect of Polchinski's proof of the no-ghost theorem. In equation 4.4.19 (at the bottom of page 140 in the latest edition) he considers the following state (or rather ...
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1answer
50 views

Can a single-qubit state be nontrivially extended to a non-pure state?

Consider a generic single-qubit state $$\rho=\lambda_1\lvert \lambda_1\rangle\!\langle \lambda_1\rvert+\lambda_2\lvert \lambda_2\rangle\!\langle \lambda_2\rvert\in\mathcal H_S.$$ I am interested in ...
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3answers
83 views

What is the Hamiltonian in the “energy basis” for a simple harmonic oscillator?

My textbook says that for a simple harmonic oscillator the Hamiltonian can be expressed in the "energy basis" in this way: $$\hat H=\hbar\omega\bigg(\hat a^{\dagger}\hat a + {1\over 2}\bigg).$$ I ...
4
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1answer
91 views

Why is time-reversal represented by an antilinear and antiunitary operator? [duplicate]

Operators related to physical transformations in quantum mechanics are usually unitary and linear except time-reversal which is both antiunitary and antilinear. What is the explanation for this ...
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2answers
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What's the time derivative of the Annihilation operator?

I've been dealing with annihilation operator recently where you can see related information Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction How to ...
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If $\sum_n \ c_n \ \psi_n(x,t)$ represents an arbitrary state for a given solution to the TISE, what are the bases for a free particle?

If $\sum_n c_n \psi_n(x,t)$ represents an arbitrary vector in the Hilbert space of solutions to Schrodinger's equation with a given potential function, this makes makes sense to me. Each $\psi_n$ can ...
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1answer
35 views

Considering an arbitrary wavefunction for a free particle, are all normalizable functions valid?

One can show that a possible solution to a wavefunction with constantly zero potential is equal to, only considering the spacial piece: $$\psi(x) = \int_{-\infty}^{\infty} A(k) \ e^{ikx} dk$$ This ...
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1answer
54 views

Is it possible to calculate the average value of $x^{2}p^{2}$ for an infinite square well? [duplicate]

If you can only measure either position and momentum in quantum mechanics how would one find the average value of $x^{2}p^{2}$ for an infinite square well?
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Quantum Wave vs Classical Wave

The wave in quantum mechanics is not a wave in true sense of water wave. It is a mathematiccial wave. It is a probability wave. I think I get till here. The question is does this probability wave (for ...
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1answer
77 views

Statevector formalism,$|\psi\rangle =c_1|A\rangle +c_2|B\rangle \neq (c_1a_1+c_2b_1)|u_1\rangle +(c_1a_2+c_2b_2)|u_2\rangle $

In statevector formalism suppose two particle $|\psi\rangle =c_1|A\rangle +c_2|B\rangle $ where $|A\rangle =a_1|u_1\rangle +a_2|u_2\rangle , |B\rangle =b_1|u_1\rangle +b_2|u_1\rangle $, but $|\psi\...
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1answer
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Finding $T^\dagger$ where $T\varphi(x)=\varphi(x+a)$

Finding $T^\dagger$ where $T\varphi(x)=\varphi(x+a)$. It is a pretty straightforward and I imagine easy question.
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1answer
41 views

Can this wave function be normalized? [closed]

This question I am stuck on goes like this: The ground state wave function for the electron in a hydrogen atom is $c\ e^{-r/a}$ where $r$ is the radial coordinate of the electron, $c$ is a constant ...
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2answers
165 views

In quantum mechanics, why is $\langle T\rangle=\frac{\langle p^2 \rangle}{2m}$ rather than $\langle T\rangle=\frac{\langle p \rangle^2}{2m}$?

I'm a newbie reading quantum mechanics from "Inroduction to Quantum Meachanics" by Griffiths and in the early pages of the book the author defines: $$\langle x\rangle =\int_{-\infty}^{\infty} x|\Psi(...
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1answer
28 views

Wave function - expected momentum is infinite

I have a wave function, say $$ \begin{align} \Psi(x) = xe^{ikx} \end{align} $$ And I want to find the expectation value for momentum, $<p>$ It's not working out to an integral I can evaluate ...
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2answers
56 views

Orthogonal wave functions

I am wondering: can we explain the concept of orthogonality in physics for a beginner (without much math and linear algebra) by saying it simply means that the particle can not exist in two different ...
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1answer
207 views

When can one find a canonically conjugate operator?

Suppose one is given a self-adjoint operator $A$ acting on an infinite dimensional separable Hilbert space $\mathcal{H}$. Under what conditions can one find an operator $B$ such that $[A,B] = i$? And ...
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1answer
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Is there a function which is square integrable and doesn't tend to zero at infinity but it belongs in the domain of the momentum operator?

Is there a function which is square integrable and doesn't tend to zero at infinity but it belongs in the domain of the momentum operator? There are some counterexample for functions that are square-...
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160 views

What is Hilbert Space? [closed]

What is Hilbert space? Please explain in it in very easy English. I've seen so many answers here that are so easy to understand but the language itself makes it hard to understand. That is why I have ...
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1answer
52 views

Question regarding position and momentum representations

According to Cohen-Tannoudji's Quantum Mechanics book we can pick the following two bases composed by functions that doesn't belong to $\mathscr{F}\in L^2(\mathbb{R^3})$: $$ \xi_{\mathbf{r}_{0}}(\...
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0answers
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Priority of tensor product and inner product and outer product

In order to find the following calculation: σ x ( ( | ψ ⟩ ⊗ | ϕ ⟩ ) ( |...
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1answer
60 views

How to generate ladder operators for an arbitrary Hamiltonian?

How to generate ladder operators for an arbitrary Hamiltonian? i.e. for a power-law potential.
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1answer
47 views

Proof of strong convexity of trace distance

I'm trying to follow the Nielsen and Chuang proof (equation 9.49 of Chapter 9, page 408). I reproduce it here for completeness. With trace distance defined as $D(\rho, \sigma) = \frac{1}{2}tr(|\rho - ...
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1answer
38 views

Probability of $\frac{-1}{\sqrt2} S_x + S_z $

I have a State $\left|\Psi\right>=\frac{\left|1\right>+\left|0\right>}{\sqrt{2}},$ in the $z$-Spin basis and want to calculate the probability of this state for the eigenvectors of the ...
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Dimension of Hilbert space descriptions of complementary variables

Hilbert space and its different sets of orthonormal bases (ONBs) provide an excellent framework for describing complementarity. In spin-1/2 space we chose the $(z^+,z^-)$ basis and make measurements. ...