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

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2answers
43 views

When generalizing from discrete (but infinite) eigenstates to continuous eigenstates, Why do we change the definition?

The propagator function for discrete eigenstates is $$u(t)=\sum_{n=1}^{\infty}|E_n\rangle\langle E_n|e^{-iE_nt/ \hbar } \tag{1}\ .$$ But when we have continuous eigenstates, (like for the case of ...
0
votes
1answer
70 views

Why tensor product? [duplicate]

Let $A$ an $B$ be two discrete observables (like spins). When exactly and why we have to consider their tensor product when talking about the mutual observation of the corresponding phenomena?
-5
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0answers
83 views

Funny quantum joke [duplicate]

Ok guys, this should be a fuzzy/silly question, but I have to understand why we do that (id est: the sign meaning). Let's suppose I want to describe, as a joke, the classical state of a coffee ...
4
votes
2answers
136 views

How can mean value of a quantity $be$ an operator?

In Laundau & Lifshitz Quantum Mechanics. Non-relativistic theory in $\S29$ a problem is given: PROBLEM Average the tensor $n_in_k-\frac13\delta_{ik}$ (where $\mathbf{n}$ is a unit vector along ...
0
votes
1answer
41 views

What is the relationship between completeness of wave functions and completeness of Hilbert space?

In the lecture, my prof said that completeness means that any wave function can be constructed using an infinite number of "other" basis wave functions. This is very intuitive since this is nothing ...
0
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0answers
32 views

Variational principle proof (summing over $n$)

From http://en.wikipedia.org/wiki/Variational_method_%28quantum_mechanics%29 $$= \sum_n \sum_m c_n^*c_mE_m \langle \psi_n|\psi_m \rangle$$ $$= \sum_n |c_n|^2E_n$$ I just want to better understand ...
5
votes
2answers
292 views

Can expectation value be imaginary?

I was solving a problem and the result of the expectation value of an operator came out to be $-\frac{\hbar}{4}$ $i$. Is this result possible? It seems counter intuitive.
0
votes
2answers
69 views

Calulate the eigenvalues and the possible states after measurement

An observable is given by $$\sum\limits_{n= 1}^N a_n|a_n\rangle\langle a_n | $$ Here $\langle a_n |a_m\rangle = \delta_{nm}$. What are the possible measurement results corresponding to the operator ...
2
votes
1answer
70 views

Is the Hilbert space spanned by both bound and continuous hydrogen atom eigenfunctions?

As e.g. Griffiths says (p. 103, Introduction to Quantum Mechanics, 2nd ed.), if a spectrum of a linear operator is continuous, the eigenfunctions are not normalizable, therefore it has no ...
3
votes
1answer
93 views

States in light cone string theory

Currently I'm trying to understand string theory in the light cone quantization. I just have had a look into Polchinski (Vol. 1, Introduction to the bosonic string), because – as far as I could see – ...
0
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2answers
75 views

What is the necessity of wave packet in studying matter wave?

I am new to this realm of physics. I have literally understood the matter wave, wave function; read the trapped electron in an infinite potential-well. But what I didn't understand is the concept of ...
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0answers
38 views

Creating a Hermitian function [migrated]

Say I have an operator $A$ such that $A^\dagger = B$. I want to construct a Hermitian function, $f$, of these operators, $f(A,B)^\dagger = f(A,B)$. Is it possible to construct a function $f$ such that ...
4
votes
4answers
586 views

What is the difference between + and - signs in superpositions of quantum states?

What is the difference between states $$ \frac1{\sqrt{2}} |11\rangle+\frac1{\sqrt{2}} |00\rangle $$ and $$ \frac1{\sqrt{2}} |11\rangle- \frac1{\sqrt{2}} |00\rangle~? $$ They will all eventually ...
4
votes
2answers
145 views

Isomorphism of rigged Hilbert spaces

In connection with the statement that QM can be formulated in terms of separable complex (rigged) Hilbert spaces, the fact that all infinite dimensional separable complex Hilbert spaces are isomorphic ...
1
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0answers
22 views

representation of spinors

I am trying to get from the abstract representation of Spinors, as wave functions $|\Psi \rangle$ in the base of tensors products $| S_z \rangle \otimes | x \rangle$ of eigenvectors of the spin ...
3
votes
2answers
96 views

How to associate a Hilbert space with a QM system?

I couldn't really find a fitting title for this question. I'm still relatively new to QM and am trying to get the basics down. I understand that a physical system is associated with a Hilbert Space, ...
0
votes
1answer
32 views

A series of bound states covering an interval

Generally, the bound states (normalizable eigenvectors) of a Hamiltonian have discrete eigenvalues. Is it possible for the eigenvalues to cover an interval? Say, $(a,b)$? That is, for each $E \in ...
1
vote
1answer
97 views

About the orthogonality of the Hamiltonian eigenstates for the the continuous energy spectrum

I would like first to describe a strange case that I encountered. $ \ \ - $ I solved the Schrodinger equation with a potential barrier (a potential well limited by a finite height wall which decrease ...
1
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1answer
39 views

Regarding calculations with plane waves

I'm dealing with some basic calculations with plane waves and I'm having some trouble with an idea. It has been said in another question that if you take to momenta like, for example ...
3
votes
4answers
118 views

Projection of wavefunction onto basis function

I am given to believe that one way that one would could represent a wavefunction is by the expansion $$\Psi(x) = \Sigma_n \Psi_n(x) = \Sigma_n f_n\phi_n(x) \tag{1}$$ where $\{\phi_n (x) \}$ is an ...
1
vote
2answers
45 views

Expectation value with plane waves [duplicate]

Hey guys Im a little confused with the concept of plane waves and how to perform an expectation value. Let me show you by an example. Suppose you have a wave function of the form ...
20
votes
3answers
686 views

Why does non-commutativity in quantum mechanics require us to use Hilbert spaces?

I am reading Why we do quantum mechanics on Hilbert spaces by Armin Scrinzi. He says on page 13: What is new in quantum mechanics is non-commutativity. For handling this, the Hilbert space ...
1
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0answers
63 views

Non Hermitian Quantum Mechanics

I was just reading about Non-Hermitian Quantum Mechanics dealing with Hamiltonians $H$ that are not Hermitian operators. Then it is unclear that we get orthonormal eigenstates. Now, I was reading a ...
2
votes
1answer
74 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
191 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
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0answers
48 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
66 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
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2answers
75 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
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1answer
35 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
168 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
99 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
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3answers
111 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
106 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
108 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
433 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
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0answers
67 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
116 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
124 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
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1answer
42 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
42 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
87 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
68 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
81 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
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1answer
30 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
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0answers
34 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
100 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
245 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
164 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
82 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 ...