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

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2
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1answer
43 views

Are these two spin states the same?

Consider two sets of axes, $xyz$ and $x'y'z'$, and the two spin states \begin{align} |\psi\rangle &= A(|+_x\rangle + |+_y\rangle + |+_z\rangle)\\ |\psi'\rangle &= A(|+_{x'}\rangle + ...
2
votes
6answers
159 views

Is $∣1 \rangle$ an abuse of notation?

In introductory quantum mechanics it is always said that $∣ \rangle$ is nothing but a notation. For example, we can denote the state $\vec \psi$ as $∣\psi \rangle$. In other words, the little arrow ...
4
votes
0answers
35 views

Derivation of the Lippmann-Schwinger equation

I was trying to understand the derivation of the Lippmann-Schwinger equation in Sakurai's Modern Quantum Mechanics, Section 6.1. Our teacher presented a much simpler derivation, similar to that on ...
0
votes
1answer
64 views

What's the meaning of the propagator in QM?

Yesterday I was solving some exercises, and after solving the time evolution I was asked to find the probability of the system to some state. In specific: $$|\Psi(t)\rangle = ...
1
vote
2answers
62 views

Bra-ket of products

I was trying to solve the following problem. (Lifted from Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory by Szabo and Ostlund) I came across across a solution for ...
1
vote
1answer
34 views

How does the Hermiticity of an operator imply that functions have an expansion in in multiple bases?

In Shankar QM it is stated that since the $\boldsymbol K$ operator is Hermitian, vectors, which are expanded in the $\boldsymbol X$ basis with components $f(x) = \langle x | f \rangle$, must have an ...
2
votes
3answers
154 views

Do we need an orthonormal basis in Quantum Mechanics?

I was wondering if it is important in Quantum Mechanics to deal with operators that have an orthonormal basis of eigenstates? Imagine that we would have an operator (finite-dimensional) acting on a ...
0
votes
3answers
83 views

What is the 'normal/standard' formulation of quantum mechanics called?

I know of at least three equivalent formulations of QM: The "normal/standard" one, dealing with Hilbert spaces and state vectors. The Feynman path-integral formulation. The Wigner-Weyl phase space ...
1
vote
3answers
106 views

Does quantum mechanics only deal with Hermitian/self-adjoint operators?

Whatever matrix I am seeing in quantum mechanics all all Hermitian matrices. We are using their eigenvalues for different types of work. Fortunately all their eigenvalues are real. Have you ever seen ...
2
votes
4answers
92 views

Physical meaning of linear combination of possible states in infinite well

The solution of infinite well, positioned from $x=0$ $x=l$, is $$ \Psi_n(x,t)= \sqrt{\frac{2}{l}}\sin\left(\frac{n\pi}{l}x\right)e^{iE_nt} $$ But the most general solution of this problem is : $$ ...
1
vote
3answers
98 views

The Hermitian operator can have many outer product representations?

Let $H$ be a Hermitian operator, so it can be written as $$H=\lambda_1P_1+\lambda_2P_2........\lambda_kP_k,$$ where $\lambda_i$ are eigen values and $P_i$ corresponding projector operators for the ...
7
votes
5answers
398 views

Hilbert space vs. Projective Hilbert space

Hilbert space and rays: In a very general sense, we say that quantum states of a quantum mechanical system correspond to rays in the Hilbert space $\mathcal{H}$, such that for any $c∈ℂ$ the state ...
0
votes
0answers
42 views

Calculating the square of angular momentum operator in cylindrical coordinates

I want to evaluate the square of angular momentum, $L^2$, in cylindrical coordinates. I found components of $L$ in cylindrical coordinate. How can I find eigenvalue and eigenfunction of $L_z$?
2
votes
1answer
108 views

Is there a simple way of finding the eigenstates of the creation and annihilation operator in QM?

How can I find the eigenstates of creation and annihilation operator in QM? My attempt: Such eigenstate will obey: $$ a^{\dagger} |\psi \rangle = \alpha |\psi \rangle. $$ We can expand $|\psi ...
0
votes
2answers
121 views

QM: operators like $\hat{\mathbf{r}} \cdot \hat{{\mathbf{p}}}$

How would we treat an operator of the form $ \hat{\mathbf{A}} \propto \hat{\mathbf{r}} \cdot \hat{{\mathbf{p}}} $ ? Would it have eigenstates that are also eigenfunctions of position and/or momentum? ...
1
vote
1answer
38 views

Wigner's Theorem and discrete Symmetries

According to my skript: A pure state is a ray: $\quad$ $\{λψ\}$, where $ψ ∈ \mathcal H$, $||ψ|| =1$ fixed and $λ ∈ \mathbb C$, $|λ| = 1$. Pure states are uniquely given by 1-dimensional orthogonal ...
1
vote
1answer
39 views

Probabilities of pure states and density operators

According to my skript: A pure state is a ray: $\quad$ $\{λψ\}$, where $ψ ∈ \mathcal H$, $||ψ|| =1$ fixed and $λ ∈ \mathbb C$, $|λ| = 1$. Pure states are uniquely given by 1-dimensional orthogonal ...
1
vote
1answer
77 views

How do we normalize a delta function position space wave function? [duplicate]

I have a position space wavefunction $$\psi(x) = \delta(x-a) + \delta(x+a).$$ Now the question states to compute the following: The Fourier transform of $\psi(x)$. (Which invariably is the momentum ...
0
votes
1answer
66 views

Configuration Space And Hilbert Space For A Physicist Without Knowledge Of Analysis

I have passed calculus course, have basic knowledge of complex numbers and passed introductory linear algebra course. I am trying to study Griffith Quantum Mechanics book, but I am also checking some ...
0
votes
1answer
52 views

Why $2j+1$ number of states?

In this statement from Modern Quantum Mechanics by J.J. Sakurai: If $j$ is an integer, all $m$ values are integers; if $j$ is a half-integer, all $m$ values are half-integers. The allowed ...
0
votes
1answer
72 views

What does vector space and bra/ket space mean? [closed]

I wonder What are the similarities and dissimilarities between a vector space and bra/ket space?
0
votes
1answer
28 views

Spin Hilbert space

I'm currently doing some quantum mechanics and was able to transform my Hamilton operator to something that basically looks like this: $$ H = H_{xy} + \frac{p_z}{2M} + \alpha S_z, $$ where $H_{xy}$ ...
1
vote
0answers
32 views

How does Dirac define the representative of $\{\langle\phi\frac{d}{dq}\}\psi\rangle = \langle\phi\{\frac{d}{dq}\psi\rangle\}$

On pate 89 of Dirac's book, The Principles of Quantum Mechanics, he writes: Let us treat the linear operator $\frac{d}{dq}$ according to the general theory of linear operators of section 7. We ...
5
votes
1answer
95 views

Equivalence classes in a Hilbert space

I'm reading something about quantum information/quantum computing theory, and I've run into a wall. I know what is meant by an equivalence class and how something can be partitioned into equivalence ...
6
votes
2answers
214 views

Infinite dimensional vector spaces vs. the dual space

I just happened across this over on Math Overflow. It references the following theorem from linear algebra: A vector space has the same dimension as its dual if and only if it is finite ...
6
votes
2answers
153 views

Does the Hermitian operator $H=-\frac{d^2}{dx^2}$ have imaginary eigenvalues?

In quantum mechanics, Hermitian operators play a very important role because they possess real eigenvalues. Considering $-\frac{d^2}{dx^2}$, it is a Hermitian operator (Actually it's the simplest ...
1
vote
1answer
60 views

Heisenberg Hamiltonian for spin-spin system

I wonder how we should conclude the following Hamiltonian (I mean the 32-18 in the picture below, written in solid state physics by Ashcroft & Mermin.) for spin-spin system? (It is in chapter 32 ...
1
vote
2answers
99 views

Why is quantum mechancis is not content with symmetric operators, but wants self-adjoint operators?

A symmetric operator has only real eigenvalues and different eigenvectors corresponding to different eigenvalues are orthogonal. These are exactly what we want for a physical observable. I think ...
1
vote
1answer
87 views

Discrete vs Continuous spectra of operators [duplicate]

Why is it that if an operator $Q$ has a discrete spectra, that the eigenfunctions are all in Hilbert space? Why is it that if the spectrum is continuous we automatically know that the eigenfunctions ...
1
vote
1answer
39 views

Normalize Triplet State of Hydrogen

For hydrogen, the total spin of the electron and proton is $s = 1$, while $m_s = -1,0,1$. If $m_s = 1$, one of the states can be written as $$\left| 1\;1 \right > = \left |\uparrow \uparrow\right ...
1
vote
2answers
277 views

Hilbert space and Hamiltonians

Assume a system described by a Hamiltonian H, and assume that the eigenstates of H, $φ_i$(r) are integrable in absolute square. We say that these states belong to a Hilbert space (they can even form a ...
0
votes
1answer
41 views

Hamiltonian acting on sum operator

I am following a derivation in a book. It is implementing a state $|{\psi}\rangle$ into the eigenvalue equation $\hat{H}|{\psi}\rangle=E|{\psi}\rangle$. The $|{\psi}\rangle$ term contains a ...
0
votes
3answers
150 views

About the postulates of quantum mechanics and self-adjointness

I am a freshman trying to understand the very basics of quantum mechanics but I met barriers at the beginning. What really matters is the postulates of quantum mechanics and their relationship with ...
1
vote
1answer
60 views

Is operator $\hat{O}_{\alpha}:|\phi,\psi\rangle\mapsto |e^{i~\alpha[\phi,\psi]}~\phi,e^{-i~\alpha[\phi,\psi]}~\psi\rangle$ unitary?

Is the operator $\hat O_{\alpha}$ which is defined in the following a unitary operator? Operator $\hat O_{\alpha}$ is supposed to act on composite states with two explicit components, such that ...
1
vote
1answer
59 views

Projection operators in a direct product space

The things I'm pretty sure I understand: Let's say I have a single particle hamiltonian $H$ represented by a $2$x$2$ matrix, so it has two eigenstates $|\lambda_1\rangle$ and $|\lambda_2\rangle$. I ...
22
votes
3answers
2k views

Why do we need infinite-dimensional Hilbert spaces in physics?

I am looking for a simple way to understand why do we need infinite-dimensional Hilbert spaces in physics, and when exactly do they become neccessary: in classical, quantum, or relativistic quantum ...
1
vote
1answer
54 views

What is the relation between renormalization and self-adjoint extension?

What is the relation between renormalization and self-adjoint extension? It seems that a renormalization scheme can be rigorously treated mathematically using the self-adjoint extension theory. Is ...
1
vote
1answer
30 views

Addition of angular momenta, coefficient in the |10> state

In Griffith's text, they apply the lowering operator on the |$11\rangle$ state to get the |$10\rangle$ state. They show this result in two forms on pg. 185: $S_{-}\left(\uparrow\uparrow\right) = ...
1
vote
1answer
62 views

How is $\langle \psi |\psi \rangle$ meaningful in Dirac notation?

I'm reading Quantum Mechanics - A Modern Development, and it explains bra-ket notation, if I understand it correctly, as follows. Let $V$ be a vector space, and let $F$ be a linear function mapping ...
2
votes
2answers
34 views

String Excited States / Spectrum

I am working through David Tong's lecture notes on string theory. The bit I am stuck on is page 41/42 in Chapter 2. The PDF file is here. In 2.3.2 "The First Excited States" on page 41/42, he says: ...
2
votes
2answers
108 views

Separability of a Hilbert space and its implications for the formalism of QM

In the text I'm using for QM, one of the properties listed for Hilbert space that is a mystery to me is the property that it is separable. Quoted from text (N. Zettili: Quantum Mechanics: Concepts and ...
1
vote
3answers
103 views

State vector vs density operator

We formulate quantum mechanics using language of state vectors. One alternative formulation is possible using density operator or density matrix. Why we are doing this alternative approach? Is the ...
0
votes
2answers
95 views

Creation and annihilation operators in Hamiltonian

If I find a Hamiltonian $H = \sum_{k} \varepsilon_k a_k^{\dagger} a_k + \sum_k V_k a_k^{\dagger} a_k$ then I was wondering: As far as I know this is many body theory and so these operators act on ...
2
votes
1answer
38 views

Ambiguity in ordering of isospin states for Clebsch-Gordan coefficients

In studying isospin for nuclear physics, I am confused a bit by an ambiguity I found. If a process that goes from $K^- + p \rightarrow \Sigma^0+ \pi^0$, I can write the isospin for the left hand side ...
0
votes
1answer
50 views

Where did I go wrong in ket vector in quantum mechanics? [closed]

Consider a system of total angular momentum j = 1. The operator jx is given by: What are the possible values when measuring jx? My attempt: The eigenvalues after calculation are -1, 0, 1. Now I ...
2
votes
1answer
89 views

Probability of energy from wavefunction

In general, given a wavefunction $\psi(x)\equiv\langle x\vert\psi\rangle$ for some system, how can one compute the probability that the system will be at a given energy level $E_n$? That is, how can ...
0
votes
0answers
48 views

What lives in the Hilbert Space? [duplicate]

Consider the eigenvalue equation: $$\hat{Q}\Psi = q\Psi$$ where $q$ and $\Psi$ are eigenvalues and eigenfunctions of the hermitian operator $\hat{Q}$. If the spectrum of the hermitian operator is ...
0
votes
1answer
45 views

Integral over scalar product of eigenfunction of momentum operator and harmonic oscillator one

Recently I've met following expression: $$ \tag 2 \sum_{n}f(n)\int dp~ |\langle p | n\rangle|^{2} = 2\pi \sum_{n}f(n). $$ Here $|n>$ is eigenfunction of harmonic oscillator with energy $$E_{n} = ...
0
votes
1answer
72 views

Commutation relation of position and momentum using Dirac notation

This is likely a very trivial/silly question, but in following a derivation of the position and momentum commutation relation using the dirac notation, I am having trouble justifying a certain step. ...
2
votes
2answers
80 views

What is the physical meaning of $a_{\vec{p}} \! \mid \! 0 \rangle$

$a^\dagger_{\vec{p}} \! \mid \! 0 \rangle = \mid \! p \rangle$ is interpreted as a creation of a particle with momentum $p$ from the vacuum. $a_{\vec{p}} \! \mid \! p \rangle = \mid \! 0 \rangle$ is ...