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

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34 views

Complex vector on Block sphere [closed]

I've the following problem. Given this vector on a 3D complex space: $$\\\\ {\phi_{1}} = \begin{matrix}% 1/2(-1, & i\sqrt{2}, &1)^{T} \end{matrix}\\ $$ Is it possible to draw it on the ...
3
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0answers
39 views

How to build QM with projective spaces from the beginning?

In conventional treatment of QM, one assumes that (1) physical states are normalized vectors in (rigged) Hilbert spaces and (2) operators correspond to observables, with their eigenvectors denoting ...
1
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1answer
127 views

Kronecker sum or direct sum?

When we write $$H=\sum_k H_k$$ in condensed matter physics, are we using Kronecker sum or direct sum? I think this is direct sum. However, Wikipedia says it is Kronecker sum. Can anyone give some ...
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1answer
89 views

Clarification about two forms of the wave function

The wave function in the position representation is $\langle\ x\rvert\psi\rangle$ = $ \psi (x) $ , where $ \psi (x) $ are the continuous coefficients that multiply the orthonormal basis vectors, i.e, ...
2
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1answer
77 views

Proof that trace is independent of representation [closed]

$$\begin{align} \sum_{a'} \langle a'|X|a'\rangle &=\sum_{a',b',b''} \langle a'|b'\rangle \langle b'|X|b''\rangle\langle b''|a'\rangle \\ &=\sum_{b',b''} \langle b''|b'\rangle \langle ...
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0answers
32 views

How do calculate <p|x>? [duplicate]

In my quantum mechanics lectures it says that the relation between the basis $|x\rangle$ and $|p\rangle$ is given by: $\langle p | x \rangle = \Large \frac{e^{-ip x/ \hbar}}{\sqrt{2\pi \hbar}} \, .$ ...
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3answers
136 views

Physical Explanation of Quantum Mechanics Notation? [closed]

CLARIFICATION: I just don't understand what the notations below mean and how to use them. ============= I just started taking QM, and the new notation is quite confusing. While the math makes a ...
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0answers
58 views

Is my expansion of the state $| x \rangle$ correct? [duplicate]

In my quantum mechanics textbook it says that the relation between the basis $|x\rangle$ and $|p\rangle$ is given by: $\langle p | x \rangle = \Large \frac{e^{-ip x/ \hbar}}{\sqrt{2\pi \hbar}} \, .$ ...
2
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1answer
71 views

A question about the uniqueness of Riesz representation theorem

I am sorry this question may be too math related. However, I come from physics background and I would like to ask for an physicist's explanation. As far as I know, the Riesz representation theorem ...
0
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1answer
35 views

Heisenberg picture transition amplitudes

I want to calculate the transition amplitude for a particle to start at position $q_1$ at time $t_1$ to position $q_2$ at time $t_2$ in the Heisenberg picture. As we are in the Heisenberg picture, ...
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1answer
45 views

Why is the angular momentum added for two independent electron system? (no problem)

There is no problem now. But somebody may be confused by the same analysis when studying QM or Group theory. (actually my motivation for asking this question comes from the SU(5) Grand Unification ...
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2answers
73 views

Negative powers of operators

This may sound like a strange question, but just to be sure: Suppose I have a general Hermitian operator in Hilbert space whose action on an eigenvector is given by $R|r\rangle = r|r\rangle$. Then, I ...
2
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1answer
156 views

Hydrogen atom in superposition of energy eigenstates

Suppose a single hydrogen atom is in a superposition of energy eigenstates: $$ \psi = \frac{1}{\sqrt{2}}\psi_{100} + \frac{1}{\sqrt{2}}\psi_{200} \,.$$ Then energy will be $E = ...
2
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0answers
99 views

How to understand the relationship between these two geometrical structures?

During my study of quantum information processing, I occasionally meet two different geometrical structures: (a) The geometry of the Hilbert space of quantum state, where the superposition and ...
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0answers
20 views

For a quantum free particle, would it be possible to relate the wavefunction $a_E(E)$ in energy basis and $a_p(p)$ in momentum basis?

The energy for a free particle has continuous energy eigenvalues $E$. Let $u(E,x)$ be its energy eigenstates in position basis. Its wavefunction $\psi(x,t)$ can be expressed as \begin{align} ...
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0answers
51 views

How can we fix the constant of the energy eigenstates of a quantum free particle such that they satisfy the orthonormality condition?

For a quantum free particle, the momentum and energy eigenstates are compatible. The constants of the momentum eigenstates are fixed by their orthonormality. Similarly, how can we fix the constant for ...
3
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2answers
115 views

Quark composition of the neutral pion

I wonder why the neutral pi meson is $$ | \pi^0\rangle = \frac{1}{\sqrt{2}}\left(\vert u\overline {u}\rangle - \vert d \overline{d} \rangle \right) $$ and not $$ | \pi^0\rangle = ...
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1answer
66 views

Difference between twofold infinity and simple infinity in relation to quantum mechanics

$\newcommand{\k}[1]{\left | #1 \right\rangle }$ In his "The Principles of Quantum Mechanics", Paul Dirac states: $$c_1\k A + c_2 \k B = \k R$$ Given two states corresponding to the ket vectors ...
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3answers
164 views

Difficulty in understanding ket vectors in quantum mechanics

$\newcommand{\k}[1]{\left | #1 \right\rangle }$ Dirac in his book The Principles of Quantum mechanics states that: To proceed with the mathematical formulation of the superposition principle we ...
3
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2answers
59 views

What is $\gamma$ in the qutrit?

We know that the qubit is defined as follows $$\lvert\psi\rangle = \alpha\lvert 0\rangle + \beta\lvert 1\rangle$$ where $\alpha, \beta \in \mathbb{C}$. We can also rewrite the state of the qubit using ...
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1answer
102 views

What exactly is Schrodinger's Cat? [closed]

What exactly is Schrodinger's Cat? The little bit reading I did led me to believe that he wanted to assert the cat is dead OR alive only if you observe. What does it signify? How did it affect the way ...
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1answer
47 views

Combination of quantum numbers for a particle in a 3D box

For a second excited state, the three combination of quantum number corresponds to $$n_{1}=2,n_{2}=2,n_{3}=1$$ or $$n_{1}=2,n_{2}=1,n_{3}=2$$ or $$n_{1}=1,n_{2}=2,n_{3}=2.$$ This is from the text ...
3
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1answer
128 views

Is the energy always discrete?

In the von Neumann axioms for quantum mechanics, the first postulate states that a quantum state is a vector in a separable Hilbert space. It means it is assumed the Hilbert space has a basis with at ...
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69 views

Are there applications of $L_p$ spaces in quantum mechanics?

In quantum mechanics, there a lot of emphasis on $L^2$ spaces since Hilbert spaces describe states in quantum mechanics, so we have $$ \langle \psi | \psi \rangle = \int |\psi^2(x)|\, dx$$ Even ...
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1answer
75 views

Physical interpretation of the constant coefficient appearing in solution to the Schrodinger equation

The product solution to the Schrodinger's equation is $$\Psi_{n} \left ( x,t \right )=\psi\left ( x \right )\phi\left ( t \right )$$ By superposition, the solution becomes $$\Psi \left ( x,t ...
3
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47 views

commutation relations in terms of eigenstates scalar product

This question has caught my attention because I was unaware of the fact that the position-momentum canonical commutation relations could be derived out of the only assumption for $\langle x | ...
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0answers
34 views

Superposition of discrete level and continuum: Electron bound and free [duplicate]

Superposition between discrete states of a system is widely considered in the literature, but this system, e.g., a $H$ atom, can also have a continuum in its energy spectrum. Can the state of a ...
23
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5answers
2k views

What is a state in physics?

What is a state in physics? While reading physics, I have heard many a times a "___" system is in "____" state but the definition of a state was never provided (and googling brings me totally ...
0
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1answer
41 views

Why is the wave function an element of the function space? [closed]

The general wave function is of the form $$\Psi \left ( x,y,z,t \right )=\psi \left ( x,y,z \right )T\left ( t \right )$$ Solving via separation of variables and finding the product solutions ...
2
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2answers
114 views

How to interpret the field configuration in quantum field theory?

We often use the Fock space as the start point for our quantum field theory. In the Fock space we have definite physical meanings for the state. For example, the state $$|k_1k_2...k_n\rangle$$ ...
3
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1answer
97 views

Grassmann numbers in the dual space

I'm reading the section on Grassmann numbers in QFT for the Gifted Amateur and I'm confused by something said therein: First, they define a coherent state for fermions $\rvert \eta \rangle$ as ...
3
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2answers
124 views

Interpretation of $\langle \phi | A | \psi \rangle$ [duplicate]

If the current state of some quantum system is $| \psi \rangle$, what is the physical interpretation of $$ \langle \phi | A | \psi \rangle $$ where $|\phi\rangle$ is some other -maybe the same- ...
1
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4answers
135 views

Is there a reason why probability density is written as $\psi^*\psi$ instead of $\psi\psi^*$?

As the title states, I see $|\psi|^2$ written as $\psi^*\psi$ instead of $\psi\psi^*$. Are both correct or is there a reason behind it? As far as I'm aware, the only time I see this sort of ordering ...
1
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1answer
48 views

Second Quantization: The Identity Operator does not Commute?

Let me take the simplest possible example. Consider the fermonic Fock-space $\Lambda^*(\mathbb{C}^n)$ built out of a finite-dimensional, oriented single-particle Hilbert space $\mathbb{C}^n$ with ...
2
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1answer
117 views

In the algebraic formulation of Quantum Mechanics, how do probability amplitudes naturally arise?

In the algebraic formulation of quantum mechanics, consider $\mathcal{B}(\mathcal{H})$ as the set of all bounded operators on $\mathcal{H}$ (with involution, norm, etc.), which form a C*-algebra $C$. ...
11
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3answers
2k views

Quantum Joke (not a real joke, not a riddle)

Supposing I want to make a quantum joke, like writing this on a coffee machine: $$| \text{Status}\rangle = \frac{1}{\sqrt{2}}\ \big( | \text{Working}\rangle \color{red}{\pm} | \text{Down}\rangle ...
2
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2answers
96 views

Normalisation of free particle wavefunction

The wavefunction $\Psi(x,t)$ for a free particle is given by $$\Psi(x,t) = A e^{i(kx-\frac{\hbar k}{2m}t)}$$ This wavefunction is non-normalisable. Does this mean that free particles do not exist in ...
1
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1answer
109 views

Expectation value of an operator and commutator relation

I have a quantum operator $A.$ It's expectation value is constant respect to time. I mean $$\langle \psi(t)|A|\psi(t)\rangle$$ is a constant values. If I know $|\psi(t)\rangle$ is not an eigenstate ...
2
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1answer
93 views

Inserting the resolution of identity correctly

In a text on path integrals, I find the following: \begin{equation} \langle q_{j+1}|e^{-i(\hat{p}/2m)\delta t}|q_j\rangle = \int\frac{dp}{2\pi}\langle q_{j+1}|e^{-i(\hat{p}/2m)\delta ...
1
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1answer
82 views

Momentum in quantum harmonic oscillator with step up and step down operators [closed]

I'm hitting a wall in my understanding of the momentum operator in a quantum harmonic oscillator. I've showed that $p = (a^\dagger - a)\sqrt{\frac{m w \hbar}{2}}i$ where $a^\dagger$ and $a$ are the ...
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2answers
182 views

What is a Hilbert Space?

The Hilbert Space is the space where wavefunction live. But how would I describe it in words? Would it be something like: The infinite dimensional vector space consisting of all functions of ...
1
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2answers
70 views

Simple question: Euclidean versus Hermitian form

This may be a basic question, but why is the inner product of bra and ket Euclidean inner product [link] and not more general Hermitian form? [link] Is there something fundamental stating that $M$ ...
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1answer
41 views

Parity operator eigenstates [closed]

I have a problem I cannot solve on my own. I have given two states $\psi_1$ and $\psi_2$ and an Operator $O$ such that $P \psi_1 = \epsilon_1 \psi_2$, $P \psi_2 = \epsilon_2 \psi_2$ and $POP^{-1} = ...
1
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1answer
36 views

State-operator map, and scalar fields

Up so far, i have been studied state-operator correspondence, $i.e$, i have been questioned State operator corrponding $i.e$ $S\times S$ to $R^2$ which was wrong question. By studing Ginsparg's ...
1
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0answers
71 views

Unitary gauge transformation [closed]

I have the Hamiltonian of a charged particle without Spin in a time- independent magnetic field and an external Potential (no electric field). Also I have the standard Gauge-Transformation $A ...
3
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1answer
144 views

In nonrelativistic Quantum Mechanics, is the expectation value of a sum of operators always equal to the sum of the expectation values?

Suppose that $\lvert \psi_n \rangle$ are the eigenvectors of a Hamiltonian, $\hat{H}$, which span some Hilbert space $\mathcal{H}$ and satisfy $$\hat{H}\lvert \psi_n \rangle = E_n \lvert \psi_n ...
3
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0answers
77 views

Picture-independence of quantum mechanics

I've been thinking about the equivalence of the Heisenberg and Schrödinger pictures of quantum mechanics in the following terms lately: a quantum system is a Hilbert space $\mathcal{H}$ equipped with ...
0
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2answers
102 views

Coefficients and wavefunction in quantum mechanics

In general quantum mechanics we represent the state of a system with a state vector $| \psi \rangle $ in some Hilbert space in some base. Assuming a complete discrete set of bases vectors $ |n \rangle ...
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2answers
47 views

Probability that a measurement will be in some set

Let $\mathcal{H}$ be the Hilbert space of a quantum system and $A$ one observable in $\mathcal{H}$. If $A$ has discrete spectrum $\{a_n : n \in \mathbb{N}\}$ for simplicity, then by the postulates of ...
4
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5answers
183 views

Looking for clarification on superposition [closed]

I have always had a hard time accepting the concept of superposition from quantum mechanics. I know that the leading physicists say that the cat is both alive and dead until it is observed and that an ...