Questions tagged [hilbert-space]

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

Filter by
Sorted by
Tagged with
-1
votes
1answer
87 views

What is the simplest way to get from $\hbar$ to wave functions? [closed]

We all somehow learn that $\hbar$ is not zero in the universe. What is the simplest way to deduce (the existence of) wave functions from this observation? (Either with little math or lots of math, ...
3
votes
2answers
107 views

When determining whether a ket-vector is normalized, should the ket be complex conjugated?

The condition for normalization for a ket vector is $$\langle A \mid A\rangle = 1.$$ However, to test if ket $\mid A \rangle$ is normalized, should I form the inner product with its complex conjugate $...
1
vote
2answers
110 views

Confusion with Dirac notation in quantum mechanics

My professor was showing us how to derive the ground state wavefunction for the quantum harmonic oscillator. He begins with the annihilation operator acting on the lowest energy eigenvector: $$a|E_0\...
1
vote
2answers
29 views

Finite barrier. Constant including minus or not?

For a finite potential barrier of magnitude $V_0$ between $x=-a$ and $x=a$ we know that the time independent schrodinger equation is $\Psi'' +\frac{2m}{\hbar}E\Psi=0$ for $x<-a$. Let $E<V_0.$ ...
1
vote
2answers
58 views

How do I compute $\langle r|P|\psi\rangle$ for a given state $|\psi\rangle$ with the associated wave function $\psi(r)=\langle r|\psi\rangle$?

How do I compute $\langle r|P|\psi\rangle$ for a particle in the state $|\psi\rangle$, with the associated wave function $\psi(r)=\langle r|P|\psi\rangle$, where $P$ is the momentum operator and $r$ ...
0
votes
2answers
41 views

Normalising the exponential solution of the infinite well

I am having trouble normalising the wave function of the Schrodinger equation for the infinite well. Using the sin and cos approach I get $A=\sqrt\frac{2}{L}$, but using exponentials I get A=$\sqrt\...
0
votes
0answers
31 views

Physical meaning of quadratic forms

Let $Q$ be a dense linear subspace of the Hilbert space, $\mathcal{H}$. Then there is a mapping $q: Q \times Q \to \mathbb{C}$, known as a quadratic form. For two states $\psi, \phi \in Q$,and a self-...
2
votes
1answer
35 views

Eigenstates for a particle in a spherically symmetric potential

Consider a particle in a spherically symmetric potential. Write down a complete set of commuting observables for the system, the relevant eingenvalue equations, and the corresponding eigenstates in a ...
1
vote
0answers
47 views

Proof with complete set of eigenvectors [closed]

I have to prove, that if $\hat{\textbf{A}}$ and $\hat{\textbf{B}}$ are self-adjoint operators, and each one of them has its complete set of eigenvectors, and if $\hat{\textbf{A}}\hat{\textbf{B}} = \...
5
votes
1answer
69 views

Does QFT renormalization preserve Hilbert space?

In the Wilsonian picture, a renormalized theory is about change of scale. As we change scale for a quantum field theory, does the Hilbert space of a theory remain unchanged?
2
votes
1answer
42 views

Linear independence of a set of states, and the non-vanishing determinant of the matrix comprised of their inner products

So I have a very basic question in linear algebra, but I'll phrase it in terms of QM. Suppose we are given a set of $N$ states $\{ | \psi_i \rangle\}$. Construct the $N \times N$ matrix $$\mathcal{...
1
vote
4answers
66 views

Symmetry transformations: a doubt about the relations that we assume true

When we deal with symmetry transformations in quantum mechanics we assume true that, If before the symmetry transformation we have this $ \hat A | \phi_n \rangle = a_n|\phi_n \rangle,$ and after ...
0
votes
1answer
37 views

In QM perturbation theory, is the system generally in an eigenstate of the perturbing Hamiltonian, $\hat H_1$?

In my notes the derivation of the second order energy correction we don't do the following: $$\sum_k a_{nk}\langle\phi_n|\hat H_1|\phi_k\rangle=\sum_ka_{nk}E_k^{(1)}\langle\phi_n|\phi_k\rangle$$ ...
0
votes
1answer
23 views

Central-potential solutions to the time-independent Schrodinger equation

Am I correct in saying that the central-potential solutions to the time-independent Schrodinger equation are all of the form $R(r)\Theta(\theta)\Phi(\phi)$? If so, do $R$, $\Theta$ and $\Phi$ have any ...
1
vote
1answer
52 views

How can you subtract a value from an operator/matrix?

I'm currently following Quantum Computation and Quantum Information by Nielsen & Chuang. I'm struggling to understand the derivation of The Heisenberg Uncertainty Principle in Box 2.4 page 89. I ...
0
votes
2answers
95 views

Why is $\langle x|x'\rangle=\delta(x-x')?$ [duplicate]

Yes I have seen the explanation of why this is so in quantum mechanical textbooks. However, let's use the identity operator and do the following: $$\langle x|x'\rangle =\langle x|I|x'\rangle =\int\...
1
vote
1answer
58 views

Operator acting on bras

I need some help. Suppose, $\hat{\textbf{A}}$ and $\hat{\textbf{B}}$ are operators and $|\psi\rangle$ is any state, so that $$ \hat{\textbf{A}}|\psi\rangle=a|\psi\rangle. $$ And I wonder if this ...
-2
votes
0answers
46 views

Rigorous definition of position operator in 3D

In 1D, let $$\mathcal{D}:=\{f\in L^2(\mathbb{R};\mathbb{C})\bigg|\int_{\mathbb{R}}|x^2f(x)^2|\text{ d}x)<+\infty \}.$$ Then, $\mathcal{D}$ is a dense subspace of $L^2(\mathbb{R},\mathbb{C})$ and ...
4
votes
4answers
153 views

Time derivative in Schrödinger equation

In quantum mechanics, a system is descibed by an element $|\psi\rangle\in\mathcal{H}$, where $\mathcal{H}$ is a Hilbert space. Then on $\mathcal{H}$ (or on a dense subspace of $\mathcal{H}$), we can ...
0
votes
1answer
62 views

Deriving an identity of Lorentz group representation

I have a representation of Lorentz group on Hilbert space by following rule: $$|\alpha\rangle_{F'}=U(\Lambda)|\alpha\rangle_{F}$$ where $\Lambda $ is Lorentz transformation satisfying $$x^{\mu'}=\...
0
votes
1answer
54 views

Compute the Schmidt Decomposition of a two-qubit state

I'm trying to compute the schmidt composition of $$ |A\rangle = \frac{1}{2 \sqrt{2}}(|00\rangle + \sqrt{3}|01\rangle + \sqrt3 |10\rangle + |11\rangle) $$ I've calculated the eigenvalues to be $$ 1+\...
0
votes
2answers
51 views

Normalization in perturbation theory

When we have a system with hamiltonian $H = H_{0} + V$, we can expand the ground state wavefunction $\Psi_{0}$ using the wavefunction of the non-interacting system $\phi_{0}$, that is an eigenfunction ...
1
vote
0answers
30 views

How to prove that different squeezed vacua are the ground states of inequivalent CCR representations?

one can find on wikipedia articles on squeeze operators and squeeze coherent states these squeezed coherent states depend on a squeezed parameter r. the usual coherent states have r = 0 i have to show ...
0
votes
0answers
35 views

Stone's theorem on one-parameter unitary groups and new self-adjoint operators

I have been following the proof of the Stone's theorem on one-parameter unitary groups. The question is if the current list of self-adjoint operators used in quantum mechanics, including position, ...
1
vote
2answers
68 views

$\langle x | p \rangle = \psi_p(x)$ , What this anything to do with $\psi$ (Notation in R.Shankar)

In Principal of Quantum Mechanics R.Shankar Page 137 if we project the eigenvalue equation $$P|p\rangle = p|p\rangle$$ onto the $X $ basis,we get $$\langle x|P|p\rangle = p \langle x |p \rangle$$ ...
0
votes
2answers
83 views

Dirac expression derivation

In Quantum Mechanics, 2nd Edition by Davies & Betts on page 78 it states that there is a symmetry implied by the following Hermitian operator equation: $${\displaystyle \int \phi^{*}(A \psi)d \,\...
2
votes
1answer
65 views

Proof of normalizing the wave function

So suppose we have the wave function $\phi(x,t)$ in the context of quantum mechanics, that satisfies the Schrodinger's equation. We want to see that if we normalize this function at $t=0$ then it will ...
-1
votes
0answers
30 views

Norm of a specific operator

I am working on a problem for my Quantum Mechanics class. Given the operator $\operatorname{M}: \, L^2(0,1) \rightarrow L^2(0,1): \, f(x) \mapsto (\operatorname{M}f)(x) = m(x)f(x)$ I shall prove, ...
1
vote
0answers
29 views

On the adiabatic theorem

In the Adiabatic theorem explanation on Wikipedia it says: Diabatic process: Rapidly changing conditions prevent the system from adapting its configuration during the process, hence the spatial ...
2
votes
2answers
73 views

Explicit form of the wavefunctional

In quantum mechanics, one in principle can write down an explicit form of the corresponding wave-function. For example, $V_i$ for the $i$-th level of quantum harmonic oscillator. In QFT, the Hilbert ...
4
votes
0answers
111 views

What is density matrix in QFT?

In quantum mechanics exist fundamental object Density matrix. (See for introduction last chapter in Principles of Quantum Mechanics by David Skinner). Density matrix nesesary to describe systems even ...
0
votes
0answers
22 views

Is there any operator for the scattering problem of an elastic wave in Hilbert space?

From an algebraic perspective, there is a common shape for wave propagation that includes operators in Hilbert space. I want to know is there any operator in Hilbert space that describes the elastic ...
1
vote
0answers
59 views

Dependence of $S$-matrix on a coordinate system in QFT

The $S$-matrix is defined as follows (see e.g. Section 3.2 in Weinberg's "The quantum theory of fields"): $$S=\lim\exp(iH_0\tau)\exp(-iH(\tau-\tau_0))\exp(-iH_0\tau_0),$$ where the limit is taken when ...
1
vote
2answers
76 views

Inner product: operation between vectors from the same vector space or between vectors from a vector space and its dual space (Ex: bras and kets)?

I am taking my first steps in learning quantum mechanics and am learning about Dirac's bra-ket notation. I am trying to understand what the inner product is. My understanding so far: the inner ...
0
votes
0answers
51 views

Operator as exterior product [closed]

I had an excersise about operators. $| u_1 \rangle, | u_2 \rangle, | u_3 \rangle $ is an orthonormal basis.I had information that: $$ \hat{\textbf{B}} |u_1 \rangle = 2|u_1 \rangle, \\ \hat{\textbf{B}}...
2
votes
2answers
89 views

Trace of the Operator

I want to ask a question about the fundamental knowledge of trace of the an operator. The operator $A$ is $$A = v (G_r-G_a)$$ where v is the velocity operator of the Hamiltonian ($v=dH/dk$); $G_r$ and ...
3
votes
1answer
36 views

Orthogonality of states in the reciprocal space

I'm working on a Tight-Binding model problem, in particular, I want to prove that a particular set of eigenstates are orthogonal: $$|{\vec{k}}\rangle=\frac{1}{\sqrt{N^d}}\sum_n e^{i(\vec{k}-\vec{k}')\...
0
votes
1answer
52 views

Understanding bra-ket outer product infinite sum

I have an elementary question on clarifying the following expression: $ \sum_{n=0}^{\infty} |n+2\rangle \langle n| $ Can the term $|n+2\rangle \langle n|$ be observed as the outer product of the ...
5
votes
2answers
106 views

Do eigenfunctions of the position and momentum operators vary from one problem to another?

Now the eigenfunctions of the Hamiltonian clearly differ from one problem to another since the Hamiltonian depends on the potential and hence for a different potential we get a different eigenvalue ...
13
votes
2answers
206 views

Countable basis of coherent states used to express coherent states

Let $|\alpha \rangle$ be coherent state in Fock space. According to the paper "Coherent-state representation for the photon density operator" by Cahill (Phys. Rev. 138, B1566 (1965), §VII), every ...
1
vote
0answers
41 views

Essential self-adjointness

I'm reading "Quantum Physics" by Glimm & Jaffe. I came along a paragraph (or rather a footnote) that I want to understand better: My definition of essential self-adjointness is that there exists ...
0
votes
1answer
84 views

Exponential of number operator

I am not sure if $e^{-a^\dagger a}$ can be equal to $\left|0\right>\left<0\right|$. I found on my lecture notes that I can write it as $\sum_{jkn}\left|j\right>\left<j\right|a^{\dagger n} ...
1
vote
0answers
37 views

Measuring the observable

I am working through this quantum mechanics homework question and I am a little confused on what I am being asked to do. I have come to one of two possible answers but I don't think either is correct. ...
1
vote
2answers
142 views

Cauchy-Schwarz inequality in Shankhar's Quantum Mechanics

I'm trying to understand proof of this inequality. But I have some problems. So, Shankar starts a proof with definition a new vector $|z \rangle$: $$ |z \rangle = |v\rangle - \frac{\langle w|v \...
0
votes
1answer
85 views

Quantum Mechanics; Sakurai; Infinitesimal Translation

The following is a section from Sakurai's book "Modern Quantum mechanics" where he explains the translation operator $J$ commutation with position operator $\hat{x}$ on the subspace $|x' \rangle$: ...
2
votes
1answer
72 views

Linear algebra with Dirac notation

I have some questions about linear algebra. Let's say $\{|v_1 \rangle, |v_2 \rangle, |v_3 \rangle \}$ are orthonormal basis of the $\mathcal{V}(\mathbb{C})$. Then, let's define two vectors $$ |a \...
3
votes
0answers
51 views

On interacting QFTs with two masses $m$ and $M>2m$

hep-th/0412302v2 is an interesting paper by Shvedov about rigorous semiclassical covariant QFT. Shvedov talks on p.27 and p.30 about ''well-known'' properties of a putative interacting QFT with two ...
8
votes
2answers
484 views

Schrödinger equation on operated states

I understand that we can apply the Schrödinger equation to any wavefunction. Now, my question is, can we apply it to states that are being operated upon? Because, when we apply an operator on a state, ...
3
votes
2answers
58 views

Matrix representation of the operator $\hat{S}_x$ in the standard basis

I have recently been introduced to the idea of spectral decomposition of spin angular momentum operators in Quantum-Mechanics. Out of curiosity I was wondering if the the spin angular momentum ...
0
votes
1answer
24 views

Showing the energy of states is positive in SUSY

When we consider the energy of any state in supersymmetry we calculate: $$ \langle\phi|\{Q^I_\alpha,\bar{Q}^I_{\dot{\alpha}}\}|\phi\rangle=2\sigma^\mu_{\alpha\dot{\alpha}}\langle\phi|P_\mu|\phi\rangle\...

1
2 3 4 5
50