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

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45 views

Physical meaning of weight function in inner product in Quantum Mechanics

When taking the inner product of say two functions in Quantum Mechanics,we include a weight function w(x,y,z) that is usually equal to unity(in my undergraduate introductory QM course anyway). But ...
-1
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1answer
36 views

Orbital angular momentum eigenstates in the $|\mathbf{r}\rangle$ representation

Consider the orbital angular momentum operators $L^2$ and $L_z$. In the $|\mathbf{r}\rangle$ representation using spherical coordinates those operators actions are given by ...
3
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1answer
63 views

Operators, Distributions and States in QFT

First of all, I will mention what I understand (pls. correct if wrong): States are vectors in the Hilbert space, to include continuous spectrum (and thus distributions), we expand this space to ...
9
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4answers
194 views

Basis independence in Quantum Mechanics

The idea that the state of a system does not depend on the basis that we choose to represent it in, has always puzzled me. Physically I can imagine that the basis ought to just yield an equivalent ...
5
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1answer
66 views

Can I *always* decompose a normalizable function into the discrete Hydrogen spectrum?

This question has been bothering me for a while now: can one reconstruct an arbitrary (normalizable) function $\phi(\mathbf r)$ in $\mathbb R^3$, with only the discrete set of Hydrogen wavefunctions ...
1
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3answers
138 views

Momentum operator representation

If $\hat{p}$ acts on position eigenstate, it is $$\tag{1}\hat{p}\left|x\right\rangle=+i\hbar\frac{\partial }{\partial x}\left|x\right\rangle .$$ But in general $$\tag{2}\hat{p} = -i\hbar ...
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0answers
29 views

QFT: Limits in Time Ordered Correlation Function Derivation

Background In part of the derivation for the time ordered correlation function I have the following equation (This equation I am fine with - it is what follows that I am not), $$ ...
3
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2answers
133 views

Confusion with Weinberg's QFT book, volume 1, chapter 3: time translation and Heisenberg picture

Sorry if this is a naive question, but I am new to QFT. In the treatment of scattering in section 3.1 of The quantum theory of fields, vol.1, Weinberg first presented the general transformation rule ...
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0answers
53 views

QFT: Ground State Momentum - Normalisation of States

In my notes I have, $$ \left\langle \mathbf{p} \left| \mathbf{q} \right.\right\rangle = \left\langle 0 \left| {a(\mathbf{p})}\ {a(\mathbf{q})}^{\dagger} \right| 0 \right\rangle $$ I am not sure how ...
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1answer
51 views

Quantum bases conversion ($S_x$, $S_y$, $S_z$)

As part of several of my homework problems on the subject, I've had to convert between bases, for instance $|+\mathbf{x}\rangle$ in the $S_z$ basis $\left( \frac{1}{\sqrt{2}}\left( |+\mathbf{z}\rangle ...
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0answers
39 views

What is the meaning of “closure is lost” for a set of kets (or any members of a vector space)?

This is the closure relation in Quantum Mechanics: $$\sum_i |i\rangle \langle i| = 1 $$ which I understand as "the sum of the projections onto the basis vectors leaves the projected vector ...
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2answers
79 views

Manipulation of operators in quantum mechanics

I'm reading some notes on quantum mechanics that state the following. $$\langle x\rvert \left( \hat{x} + \frac{i\hat{p}}{m\omega}\right) \lvert E \rangle = 0 \Rightarrow \left( x+ ...
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1answer
97 views

Hilbert Schmidt inner product

I am desperately trying to solve the following problem, and would really appreciate help! Suppose $R$ and $Q$ are two quantum systems with the same Hilbert space $\mathcal{H}$ with ...
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1answer
26 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
32 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
119 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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2answers
79 views

Is superposition just quantum field? [closed]

A quantum particle is always in superposition state until it is measured, does it means that until we have a disturbance/excitation in the whatever quantum field by measurement/interaction the quantum ...
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1answer
86 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, ...
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1answer
63 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
133 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
55 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
63 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
30 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
66 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 ...
1
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1answer
114 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
96 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 ...
0
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0answers
19 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} ...
0
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0answers
49 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
97 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
53 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
158 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
48 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 ...
0
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1answer
83 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 ...
0
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1answer
31 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
111 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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0answers
66 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
71 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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0answers
45 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
votes
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
99 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
88 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
109 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- ...
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4answers
125 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 ...
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1answer
42 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
113 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 ...