# Tagged Questions

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

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### Transformations of states in quantum mechanics

In Classical Mechanics we usually describe the possible configurations of a system by points on a smooth manifold $M$ which is the configuration manifold of the system. In that case, when we talk ...
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### Does every Hilbert Space carry a representation of Poincare group?

We know all infinite dimensional Hilbert Spaces are unitarily equivalent. It should follow therefore that if I have an unitary representation of say Lorentz or Poincare group on one infinite ...
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### Why are eigenspaces of a Hermitian operator mutually orthogonal? [closed]

In Quantum Mechanics, from the properties of the solution of Schrodinger's Equation inside the infinite well, is that they are: Mutually orthogonal for different eigenvalues. Orthonormal. Complete. ...
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### General formula for expanding wave function in terms of orthogonal states?

Given a wave function $\psi(x) = \langle \psi | x \rangle$. It can be expanded in terms of orthogonal states: $$\langle \psi | x \rangle = \sum_n \langle \psi | n \rangle \langle n |x \rangle$$ ...
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### Unitary Transfomation from One Basis to Another [closed]

So we have two orthonormal linearly independent basis $\{ |\phi_1 \rangle, \dots, |\phi_n \rangle \}$ and $\{ |\psi_1 \rangle, \dots, |\psi_n \rangle \}$. We can express the basis vectors of the ...
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### Quantum States, Hilbert Space and Time

I'm having troubles with the assertion "(normalizable) wave-functions constitutes (projective) Hilbert space". The standard argument I find for this seems to go as following: say $\Psi(\vec{x},t)$ is ...
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### Proportionality of states in quantum harmonic oscillator

What is the justification for $a_{\pm} \psi_{n}$ being proportional to $\psi_{n\pm1}$ in a quantum harmonic oscillator? Here $a_{\pm}$ is the raising/lowering ladder operator.
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### Why can differentiating a function carry it out from Hilbert space?

I was just doing a QM Griffiths Problem. I was able to get it correct, but I have a few questions. Let $f(x)=x^v$ be defined on $[0,1]$ where $v= 1/2$ then $df/dx = \frac{x^{-1}}2$ Then, we know ...
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### Proving polarization identity for operators in complex vector spaces

In general for two operators to be equal, all their (matrix) elements must be equal $$A = B \rightarrow \langle \phi_1|A| \phi_2\rangle=\langle \phi_1|B| \phi_2\rangle$$ However, I am asked to show ...
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### $(H\Psi(x,t))^*=H\Psi^*(x,t)$?

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 ...
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### 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 ...
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### 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 $$\psi_{sp^3}= c_1\psi_{2s}+ c_2\psi_{2p_{x}} + c_3\psi_{2p_y}+ c_4\psi_{2p_{z}}$$ ...
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### 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 ...
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### 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 states,...
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### 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 ...
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### 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 ...
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### 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 ...
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### Problem in understanding Feynman's explanation of the Dirac-Delta function [duplicate]

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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $O = \sum_j \lambda_j P_j$...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 wrong)...
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### 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 ...
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### 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 ...
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### 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 ...
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 $$\tag{1} \langle x_f, ... 2answers 96 views ### Is there an intuitive explanation to the fact that the solutions to the time-independent Schrödinger equation form a complete basis? We were always told that the solutions to the time-independent Schrödinger equation form a complete eigenbasis for the space of all functions (all functions?) but I never understood why this is the ... 2answers 140 views ### Time evolution in Quantum Mechanics abstract state space As I've learned the first postulate from Quantum Mechanics can be stated as follows: The states of a quantum system are described by vectors in a complex Hilbert space \mathcal{H}. The book ... 3answers 566 views ### Physical meaning of quantum operators Let's say we have a wavefunction \psi and a measurement operator \hat A. I understand how eigenvalues and eigenvectors of \hat A describe the possible outcomes of the measurement. I also ... 3answers 756 views ### Eigenvalue for the creation operator for a coherent state [closed] For a coherent state$$ |\alpha\rangle=e^{-\frac{|\alpha|^{2}}{2}}\sum_{n}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle $$I can't solve the eigenvalue problem for \hat{a}^{\dagger}|\alpha\rangle where \... 0answers 58 views ### Ordering of eigenstates in the quantum adiabatic theorem Suppose we have an 'initial' Hamiltonian H_{i} whose eigenvalues are all non-degenerate, which we order as follows:  E^{0}_{i} < E^{1}_{i} < \dots < E^{N-2}_{i} < E^{N-1}_{i} Suppose ... 2answers 55 views ### Expectation value of S_x [closed]$$\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 \...
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 $a=Tr(M)$, $p_i = Tr(M\sigma_i)$...