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

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130 views

$(H\Psi(x,t))^*=H\Psi^*(x,t)$?

In the solutions of an exercise I got confused about the following equality $$(H\Psi(x,t))^*=H\Psi^*(x,t).$$ Is this true in general? Or in special cases? It seems to imply that H is a real matrix ...
6
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2answers
247 views

What is the connection between Hilbert Space and path integrals?

Given a space of states $|\rangle$, $|x\rangle$, $|x,y\rangle$, with the creation operators such as $\hat{\phi}(x)|y,z\rangle=|x,y,z\rangle$ for creating a particle at position $x$ and so on. How ...
0
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2answers
129 views

Why/How do the coefficients associated with atomic orbitals superposed to form hybrid orbital determine their spatial orientation?

In my previous Phys.SE question, I asked for why $ \newcommand{\k}[2]{\langle #1|#2 \rangle} c_1,c_2,c_3,c_4$ in $$\psi_{sp^3}= c_1\psi_{2s}+ c_2\psi_{2p_{x}} + c_3\psi_{2p_y}+ c_4\psi_{2p_{z}}$$ ...
1
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1answer
106 views

Matrix represenation of total angular momentum operator

I see that for total ket in QM of hydrogen atom we define a tensor product of kets of spatial and spin spaces, upon which spatial and spin operators, operate respectively. For the total angular ...
1
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2answers
127 views

Does this quote from my textbook imply that not all states are superpositions?

I read this at a book; The difference between bits and qubits is that a qubit can be in a state other than $|0\rangle$ or $|1\rangle$. It is also possible to form linear combinations of ...
1
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2answers
341 views

Problem in understanding the concept of 'superposition' as explained by Dirac

The concept of 'superposition' has really made me insane, actually. What I thought it was just simple superposition of matter waves. For instance, let's take the Double-Slit experiment: take the ...
0
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2answers
109 views

Relationship between nodes in wavefunction and orthogonality

I read that if I want to construct a wavefunction orthogonal to given $n$ orthogonal wavefunctions, then the new wavefunction should have $n$ nodes. Is this valid under all conditions? Is there a ...
0
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3answers
421 views

Is the superposition of stationary states a stationary state? If not, then why not?

I am a beginner in Quantum mechanics and as I understand,the superposition of stationary states is also a solution of time-independent Schrödinger equation (TISE). The wave functions that are the ...
0
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0answers
101 views

Problem in understanding Feynman's explanation of the Dirac-Delta function

This is quoted from Feynman's Lectures' Normalization of the states in $x$: We return now to the discussion of the modifications of our basic equations which are required when we are dealing with ...
6
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1answer
299 views

Must bounded operators have normalisable eigenfunctions and discrete eigenvalues?

When we have bound states, to my knowledge, we have states that are normalisable and a discrete energy spectrum. However, in the case of scattering states that have a continuous energy spectrum, the ...
3
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1answer
72 views

Hilbert space, does $|r\rangle$ satisfy $\langle r |r\rangle = 1$?

Let's say we start with no particles: $\mid0\rangle$. We have $\vec{p}\vert0\rangle = 0$, $H\vert0\rangle = 0$, where we are ignoring $\infty$ vacuum energy. Also, $a(\vec{k})\vert0\rangle = 0$ for ...
2
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3answers
136 views

What are the functions of these coefficients $c_1,c_2,c_3,c_4$ in $ \psi_{sp^3}= c_1\psi_{2s}+ c_2\psi_{2p_{x}} + c_3\psi_{2p_y}+ c_4\psi_{2p_{z}}$?

Hybridised orbitals are linear combinations of atomic orbitals of same or nearly-same energies. Atomic orbitals interfere constructively or destructively to give rise to a new orbital which is what we ...
1
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1answer
91 views

Decompose a Hermitian Operator into Eigenvalues and Projectors

Quantum Computing - A Gentle Introduction by Eleanor Rieffell and Wolfgang Polak states on p57 : Any Hermitian operator $O$ with eigenvalues $\lambda_j$ can be written as $O = \sum_j \lambda_j ...
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1answer
69 views

What is the time evolution operator in quantum mechanics [duplicate]

I'm curious about what happen to a system when the configuration of the system changes. If we have a system in a state $|\psi_{\textrm{in}}\rangle$ and we change the configuration of the system, the ...
1
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4answers
440 views

Trying to understand inner product notation

I I'm taking a QM course and I'm trying to make sense of why observables are sometimes conjugated for no apparent reason in their inner products. Right now I'm watching Dr. Susskind's lecture on ...
11
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2answers
422 views

How to describe time evolution in relativistic QFT?

I must confess that I'm still confused about the question of time evolution in relativistic quantum field theory (RQFT). From symmetry arguments, from the representation of the Poincare group through ...
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2answers
311 views

Probability to find a particle in a particlar state $\psi_{n}$ [closed]

I have a problem to understand the probabilities in QM. In particular, if I have a particle in state $\psi_{n}$, then we change the system and we ask for the probability to find the particle in a ...
5
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1answer
185 views

How to make rigorous the idea of a continuous complete set?

In Quantum Mechanics, when using Dirac's formalism one of its features is the expansion of state vectors into continuous basis of eigenvectors of unbounded self-adjoint operators. Let $\mathcal{H}$ be ...
3
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1answer
56 views

When can a Hilbert space with a given Hamiltonian be decomposed into non-interacting tensor product factors?

Let's say I have a Hilbert space $\mathcal{H}$ (either finite-dimensional or with a countably infinite basis) with a specified Hamiltonian $\hat{H}$, representing some quantum system. Under what ...
2
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2answers
144 views

How to prove $\mathrm{Tr}\left[|\alpha\rangle\langle\alpha|\hat{A}\right]=\langle\alpha|\hat{A}|\alpha\rangle$

For a coherent state $|\alpha\rangle=e^{-\frac{|\alpha|^2}{2}}\sum\frac{\alpha^n}{\sqrt{n!}}|n\rangle$, please show me how to prove, $$ ...
3
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3answers
429 views

Hilbert space of a quantum system

The first postulate of Quantum Mechanics as I've learned can be stated as: The states of a quantum system can be described by vectors in a Hilbert space. I've seem also some people also ...
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2answers
114 views

Commutator of position and momentum

I'm reading Sakurai's Quantum Mechanics. One of the problem in the book asks to use the relation $$ \langle{x}|p\rangle=\frac{1}{\sqrt{2\pi\hbar}}e^{\frac{ipx}{\hbar}} $$ to evaluate ...
2
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1answer
200 views

Superposition of two wave functions of different Hilbert spaces

I am trying to think of this problem for quite some time. Let's say, we have two sets of wave functions $\lbrace|\psi\rangle\rbrace$ and $\lbrace|\phi \rangle\rbrace$ and they belong to two different ...
0
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2answers
316 views

Hermitian conjugate of differential operator

Help me find $\hat{B^\dagger}$, when we know that $$\hat{B}=i\frac{d}{dr}$$ with the condition that $\hat{B}$ is defined in spherical coordinates. My approach: $$ ...
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0answers
111 views

Field operator eigenstate vs. single-particle state

I would like to make sure I understand some basic QFT. My understanding so far is that field operators measure field intensity and their Fourier transform measure intensity of field oscillation. In ...
1
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1answer
90 views

Are the position eigenkets $\lvert x \rangle$ really a basis for the space of states?

In my current understanding, matrix formulation and wave-function formulation of QM are basically the same because $\left|\psi\right>$ and $\psi(x)$ are really the same mathematical object: A ...
3
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2answers
212 views

How the position operator and the position basis are correctly defined?

In Quantum Mechanics, if one deals with wave functions, the Hilbert space in question is $L^2(\mathbb{R}^n)$ for a particle in $n$-dimensions, and the position operator corresponding to the $i$-th ...
4
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0answers
119 views

Pole in reflection/transmission coefficient and bound states

I was working on a scattering problem in a quantum mechanical system with Hamiltonian $$H_1=A^{\dagger}A=(-\partial_x+W(x))((\partial_x+W(x))).$$ One can show that a 'supersymmetric' partner to this ...
3
votes
1answer
92 views

State time evolution of a quantum harmonic oscillator with a Dirac-Delta function as an initial state [closed]

I have a question just like this Phys.SE problem here with a difference that our system is a harmonic oscillator (rather than a free particle). A particle with mass $m$ is connected to a string with ...
2
votes
1answer
89 views

Physical interpretation of the creation operators in string theory?

Is there any way to describe phsycially which each creation operator $a^{(i)+}_{n}$ in string theory does to the ground state string? Here would be my guess (although it is likely to be totally ...
4
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1answer
158 views

BRST quantization and norm

States with definite ghost number have zero norm (since ghost number is anti-hermitian and has real eigenvalues). E.G. when quantizing relativistic point particle, physical spectrum turns out to ...
5
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2answers
405 views

Why don't non-Hermitian operators with all real-eigenvalues correspond to observables? [duplicate]

Suppose you could construct an operator that was non-Hermitian but had all real eigenvalues or could at least be restricted in a way to create only real eigenvalues, why would this operator not ...
2
votes
1answer
90 views

When do two operators act on the same Hilbert space?

Suppose I want to represent the quantum state of a spinless particle. To do so, I employ a Hilbert space $\mathcal{H}_X$, which is an infinite-dimensional Hilbert space equipped with a position ...
2
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1answer
77 views

Probability amplitude for motion from $x_i$ to $x_f$ in Heisenberg picture

In M. Nakahara's book Geometry, Topology and Physics on page 19, the probability amplitude for a particle to move from $x_i$ at time $t_i$ to $x_f$ at time $t_f$ is given as $$ \tag{1} \langle x_f, ...
3
votes
2answers
94 views

Is there an intuitive explanation to the fact that the solutions to the time-independent Schrödinger equation form a complete basis?

We were always told that the solutions to the time-independent Schrödinger equation form a complete eigenbasis for the space of all functions (all functions?) but I never understood why this is the ...
2
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2answers
133 views

Time evolution in Quantum Mechanics abstract state space

As I've learned the first postulate from Quantum Mechanics can be stated as follows: The states of a quantum system are described by vectors in a complex Hilbert space $\mathcal{H}$. The book ...
6
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3answers
529 views

Physical meaning of quantum operators

Let's say we have a wavefunction $\psi$ and a measurement operator $\hat A$. I understand how eigenvalues and eigenvectors of $\hat A$ describe the possible outcomes of the measurement. I also ...
3
votes
3answers
634 views

Eigenvalue for the creation operator for a coherent state [closed]

For a coherent state $$ |\alpha\rangle=e^{-\frac{|\alpha|^{2}}{2}}\sum_{n}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle $$ I can't solve the eigenvalue problem for $\hat{a}^{\dagger}|\alpha\rangle$ where ...
2
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0answers
56 views

Ordering of eigenstates in the quantum adiabatic theorem

Suppose we have an 'initial' Hamiltonian $H_{i}$ whose eigenvalues are all non-degenerate, which we order as follows: $ E^{0}_{i} < E^{1}_{i} < \dots < E^{N-2}_{i} < E^{N-1}_{i}$ Suppose ...
-1
votes
2answers
54 views

Expectation value of $S_x$ [closed]

$$\chi=A\begin{bmatrix} 3i \\ 4 \end{bmatrix}$$ I am asked to evaluate the expectation value of $S_x$. I understand the equation $$\langle S_x \rangle=\frac{\hbar}{50} \begin{pmatrix} -3i & 4 ...
1
vote
2answers
356 views

Hermitian 2x2 matrix in terms of pauli matrices [closed]

In my studies, I found the following question: Show that any 2x2 hermitian matrix can be written as $$ M = \frac{1}{2}(a\mathbb{1}+\vec{p}\cdot \vec{\sigma}) $$ with $a=Tr(M)$, $p_i = ...
1
vote
2answers
292 views

Electron Spin state probability [closed]

Suppose there is a spin 1/2 particle in a state $\chi = {1 \over \sqrt{5}} \begin{bmatrix} 1\\ 2\\ \end{bmatrix} $. To determine the probability of finding the particle in a spin up($\hbar/2$) state, ...
4
votes
4answers
183 views

How to determine the observables rigorously?

In Quantum Mechanics, as I know, if a system is described by a Hilbert space $\mathcal{H}$, each physical quantity is associated with some hermitian operator $A : U\subset \mathcal{H}\to \mathcal{H}$ ...
3
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2answers
266 views

Wavefunctions in different Hilbert spaces

The state of a quantum system is represented by a wavefunction usually in some specific Hilbert space, .e.g of position, spin, momentum etc. But before deciding in which of these bases to decompose ...
0
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1answer
61 views

Possible values for $L_x$

I've a physical system with $l=1$ and I have to calculate the values I can obtain if I measure $L_x$ and their probability. I know that: the values I can obtain are $\ m=0, \pm 1$ $\displaystyle ...
2
votes
2answers
135 views

Superposition in Quantum Mechanics

First of all, let $V$ be a vector space over the field $\mathbb{F}$. It is possible then to show, by Zorn's Lemma that there is a basis for $V$. The main point is that although basis are quite ...
4
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2answers
722 views

State of a system in Quantum Mechanics and state vectors

I'm taking a course in Quantum Mechanics and there is something I'm not being able to fully understand. On more elementary courses on Quantum Mechanics I've been told that the idea of Quantum ...
1
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1answer
145 views

Completeness relations of eigenstates in the Heisenberg picture

I've been reading Srednicki's introduction to path integrals and I'm slightly unsure of the notation that he uses for the completeness relation of position eigenstates in the Heisenberg picture. In ...
3
votes
2answers
149 views

Distinction between state space and space of functions

In Quantum Mechanics a particle is described by its wave function $\Psi : \mathbb{R}^3\times \mathbb{R}\to \mathbb{C}$. In that sense, the state of a particle at time $t_0$ is characterized by a ...
1
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3answers
93 views

Is the perturbation Hamiltonian an observable?

In fine structure calculation we use the perturbation theory. The basic Hamiltonian $H_0$ is perturbed as: $H = H_0 + W$ First, the basic problem assume that $H_0$ is an observable. That allows to ...