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

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11
votes
4answers
445 views

Applying an operator to a function vs. a (ket) vector

I have a question regarding the effect of quantum mechanical operators. The definition that I'm familiar with says that an operator $A$ acts on a vector from a Hilbert space, $|\psi\rangle$, and the ...
1
vote
1answer
163 views

Inner product of position and momentum eigenkets

Let's define $\hat{q},\ \hat{p}$ the positon and momentum quantum operators, $\hat{a}$ the annihilation operator and $\hat{a}_1,\ \hat{a}_2$ with its real and imaginary part, such that $$ \hat{a} = ...
2
votes
2answers
121 views

Bloch Sphere and $SU(2) \to SO(3)$ map

For any matrix $U \in SU(2)$ there is an associated map from $S^2$ (the surface of a 3-disk) to itself defined by $\pi \circ U$, where $\pi$ is the projection map from $\mathbb{C}^2$ to $CP(1)$, that ...
1
vote
1answer
66 views

Null State Level 2 in CFT

I'm reading Cardy's notes on CFT. He states the following in section 4.3: $$\hat L_n\left(\hat L_{-2}|\phi_j\rangle-(1/g)\hat{L^2}_{-1}|\phi_j\rangle\right)=0.$$ I tried to work this out explicitly ...
0
votes
0answers
32 views

Experimental proof of the principle of superposition in QM [duplicate]

I have read that we need all operators in QM to be linear to confirm the principle of superposition which is experimentally well proven. I wonder how such an experiment could be made?
2
votes
1answer
110 views

What exactly is a coherent state and why is it interesting?

Please note that I do not have a background in physics, so if possible please refrain from a bunch of $ |x\rangle $ notations, unless clearly specifying what it symbolically means. So I have been ...
1
vote
0answers
112 views

Transition Between Position and Momentum Basis

I'm having some trouble following pages 55-56 of Sakurai's Modern Quantum Mechanics. We're trying to transfer from position space into momentum space. Here's a quote: Let us now establish the ...
1
vote
3answers
95 views

Can one construct a new operator in terms of the powers of another operator?

Suppose we have a quantum state, well described by its time-independent wave function Psi. And we have a well-defined Hermitian (self-adjoint) operator $A$. We successfully evaluate the expectation ...
3
votes
0answers
39 views

Self-adjoint extensions with 'teletransporting' boundary conditions

When choosing a self-adjoint extension of a Hamiltonian, in general one can obtain domains in which (i) the probabilities teleport* between points on the boundary and (ii) boundary conditions ...
2
votes
2answers
117 views

Ground state of a quantum mechanical system

When looking back at my courses of quantum mechanics, I noticed that assumptions about the ground state of a quantum mechanical system where rather vague and unprecise. It is always assumed that a ...
1
vote
1answer
89 views

Expectation value of an operator

Suppose we have: $$ \hat{Q}|\psi_1\rangle=q_1|\psi_1\rangle \\ \hat{Q}|\psi_2\rangle=q_2|\psi_2\rangle $$ with $q_1 \neq q_2$. Then consider the state: $$ ...
0
votes
2answers
114 views

Representations in quantum mechanics [closed]

This might be a very simple question. I just want someone to point me the right direction to understand things like this: $$ \langle x|x'\rangle=\delta(x-x') \\ \psi(x)=\langle x|\psi\rangle \\ ...
1
vote
1answer
56 views

Atom state vectors kets

An atom with two energy levels has 2 states (excited and ground), represented by kets $|e\rangle$ and $|g\rangle$ respectively. The atom has energy $\frac{1}{2}E_\theta$ when excited and ...
3
votes
2answers
78 views

Query on an operator acting on a function

I have a naive question about an operator acting on a well-behaved function. Let us say, we are talking about space translation operator acting on a function $\psi(x)$: ...
8
votes
1answer
184 views

Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction

The Schrodinger equation in Hilbert space is expressed as : $$\frac{\partial}{\partial t} \psi(t) = \frac{-i}{\hbar}H\psi(t). $$ Here $\frac{\partial}{\partial t} \psi(t) \equiv \psi'(t) \equiv\lim ...
4
votes
1answer
161 views

How are matrices used to represent quantities, and what is the meaning of a matrix?

So I'm reading this text on Quantum Mechanics, and it goes through a few chapters that I understand fairly well including probability. But then it says that all quantities, like position and energy ...
5
votes
2answers
130 views

Representation of indistinguishability in quantum mechanics

I was wondering that if particles are indistinguishable in quantum mechanics, then why do we still express their states $\left| \uparrow \downarrow \right\rangle$, as meaning particle 1 (in the first ...
4
votes
1answer
154 views

After using annihilation operator on vacuum state, why it is $0$ instead of vacuum?

For bosonic systems, why $a|0\rangle=0$ and not $a|0\rangle=|0\rangle$?
1
vote
0answers
69 views

How to make a base tranformation for a linear operator in QM? [closed]

I have 2 bases A and B with the following kets: Base A: $|a_1\rangle$ and $|a_2\rangle$ Base B: $|b_1\rangle = \frac{1}{\sqrt2} \cdot(|a_1\rangle + i\cdot|a_2\rangle)$ $|b_2\rangle = ...
1
vote
0answers
60 views

Quantum Rigid Rotor Perturbation

As the title says, I have a rigid rotor with a perturbation given below $$H=\frac{L^2}{2I}-\alpha B L_z.$$ So I know that the eigenvalues of $H$ will be $\ell(\ell+1)/2I -\alpha B m$ where $m$ is our ...
3
votes
2answers
143 views

What is the precise definition of state of a quantum system?

On studying quantum states of the hydrogen atom, I came with a question: what is the precise definition of a quantum state? Particularly, when solving the Schroedinger eq for the H using separation of ...
4
votes
1answer
106 views

Are there two different ways to express the position operator $x$ in terms of the creation and annihilation operator?

As we known, to express the position operator $x$ in terms of the creation and annihilation operator $a^{+}$ and $a$, one way is: $$x= \sqrt{\frac{\hbar}{2\mu\omega}}(a^++a);$$ $$p= ...
1
vote
2answers
133 views

Lippmann-Schwinger solution

What's wrong with this general solution of the Lippmann-Schwinger equation: $$ |\psi_k \rangle=|\phi_k \rangle+G_k V|\psi_k \rangle $$ Taking the inner product with $\langle\phi_{k'}|$ \begin{align} ...
4
votes
1answer
147 views

A question about the energy of turning on and off interaction adiabatically in QFT

I read a saying as follows: In a theory with no particles which decay and no bound states, the turning on and off of the interactions merely serves to limit the effective range of forces. In this ...
4
votes
4answers
534 views

Complex conjugate of momentum operator

Consider momentum operator representation in position space. $$\hat{p}=-i\frac{\partial}{\partial x} \,\ \text{and its eigen functions are } e^{ipx} \,\text{and} \,\ e^{-ipx}.$$ ...
2
votes
0answers
130 views

What are you studying when you study a Harmonic Oscillator in QM?

This probably is a naive question - so please forgive a self-studier. In the text I am studying, one builds a HO by placing a particle in a potential that increases quadratically from the origin. The ...
1
vote
1answer
75 views

Is there any non-hermitian operator on Hilbert Space with all real eigenvalues?

The property of hermitian is the sufficient condition for eigenvalue being real. Is there any non-hermitian operator on Hilbert Space with all real eigenvalues? If there exist, then can all ...
3
votes
2answers
151 views

Understanding Well Defined States

I am self-studying from a text in QM. Well defined states are mentioned several times. By and large these are consistent and seem to be readily apparent: states of well defined energy are basis ...
1
vote
1answer
52 views

Converting between (abstract) linear operators and their position representations

Just as we have an abstract state vector $|\psi\rangle$ and its position representation $\psi(\vec{x}) = \langle \vec{x} | \psi \rangle$, how do we transform between a linear operator, say $H$, that ...
1
vote
1answer
121 views

Wave Function Integral I need help conceptually and Mathematically

$$\int_{-\infty}^{\infty}\frac{\partial^2\bar{\psi}}{\partial{x^2}}\frac{\partial\psi}{\partial{x}}~dx.$$ I have read that this is equal to Zero. Only problem is that what I am reading about doesn't ...
0
votes
2answers
129 views

Quantum Mechanics - Observable

If $O$ represents an operator corresponding to an observable why does the following equality hold? $$\langle f(x)\, |\, O g(x)\rangle = \langle g(x) \,|\, O f(x) \rangle$$ It is used on the last ...
3
votes
1answer
302 views

Diagonalization of Hamiltonian

Typically, one way of understanding the physics of an interacting quantum system is by diagonalizing the Hamiltonian. In principle, can we always diagonalize a Hamiltonian, such that it is expressed ...
0
votes
2answers
131 views

Projection operators and their subspaces (of Hilbert space)

I've been watching Susskind's lectures on Quantum Entanglement, and something he said regarding (non-)commuting projection operators confused me. Consider two subspaces {$|a>$} and {$|b>$} of ...
1
vote
1answer
128 views

Singular wave function

Given a wavefunction, $\psi(x)$, is it possible for $\psi$ to be singular at a point? Are there any rules against a wavefunctions having any singularities? For instance if $$\psi(x) ...
2
votes
3answers
342 views

Wave function not normalizable

Does the solution of the Schrodinger equation always have to be normalizable? By normalizable I mean, given a wavefunction $\psi(x)$ $$\int_{-\infty}^{\infty}|\psi(x)|^2 dx<\infty \qquad ...
12
votes
5answers
1k views

How does the hydrogen atom know which frequencies it can emit photons at?

At university, I was shown the Schrodinger Equation, and how to solve it, including in the $1/r$ potential, modelling the hydrogen atom. And it was then asserted that the differences between the ...
1
vote
1answer
136 views

Normalisation of Linear Harmonic Oscillator - Ladder Operator Method

I was watching the following video on the harmonic oscillator using ladder operators : http://youtu.be/gRdCV9p8sAU?t=30m9s Clicking on the video above will take you to the exact point where my ...
1
vote
2answers
105 views

Quantum Mechanics: Momentum operator questions [closed]

I'm asked to determine $\hat{P}|\Psi_0\rangle$, $\langle{\hat{P}}\rangle$, and $\langle\hat{P}^2\rangle$ for $$\Psi_0(u) = \psi_0 + 2\psi_1$$ I understand how to make the matrix for $P$ in regards ...
1
vote
2answers
88 views

Unitary transformation between complete + orthonormal bases

Suppose the complete orthonormal bases $\{|\psi_n\rangle\}$ and $\{|\psi{'}_n\rangle\}$ are related by the transformation matrix $U$: $$ |\psi{'}_n\rangle = U|\psi_n\rangle \\ \langle\psi{'}_n| = ...
4
votes
1answer
122 views

Orthogonality of summed wave functions

Problem. I know that the two wave functions $\Psi_1$ and $\Psi_2$ are all normalized and orthogonal. I now want to prove that this implies that $\Psi_3=\Psi_1+\Psi_2$ is orthogonal to ...
1
vote
1answer
35 views

Uncertainty Definition QM

On my introductory course in Quantum Mechanics, the uncertainty of an operator $A$ in the state $\psi$ is defined by $$(\Delta A)^2_{\psi}=\langle(A-\langle A \rangle_{\psi})^2\rangle _{\psi}$$ I'm ...
1
vote
2answers
109 views

Quantum Expectation Values

I'm having trouble understanding the motivation for the definition of the expectation of a self adjoint operator $A$: $$\langle A \rangle _\psi=\int_{\mathbb{R}}\psi^*A\hspace{0.2cm} \psi ...
1
vote
2answers
379 views

Hilbert space in quantum mechanics

I think in quantum mechanics we assign to each system a specific Hilbert space i.e. if systems are different then their Hilbert spaces are different. Is this true? If not why? For differernt system I ...
1
vote
1answer
181 views

Quantum Mechanics - Rectangular Potential Barrier - Normalisation

I have a quick question regarding the normalisation of the wave function of a particle incident on a potential barrier specifically regarding the normalisation of the wave functions. The problem is ...
2
votes
1answer
311 views

Non-Hermitian operator with real eigenvalues?

So we know that in Quantum Mechanics we require the operators to be Hermitian, so that their eigenvalues are real ($\in \mathbb{R}$) because they correspond to observables. What about a non-Hermitian ...
0
votes
0answers
46 views

Completeness of the state space and Hilbert space [duplicate]

I am wondering just why is it supposed that quantum states lie in a Hilbert space, which mathematically requires completeness? In other words, what does completeness (defined in terms of Cauchy ...
1
vote
0answers
86 views

Momentum and position operators in Schrödinger representation

I was going through some intro notes on path integral (for QFT), and am stuck with this equation for position and momentum in Schrödinger (position) representation, $$ \hat{1} =\int ...
3
votes
3answers
428 views

Can the expectation value of the square of momentum be negative?

I've been solving a problem in quantum mechanics, and I was deriving the standard deviation of $P$, knowing that $\langle P\rangle=0$. Because $\Delta P=\sqrt{\langle P^2 \rangle - \langle P \rangle ...
0
votes
1answer
118 views

Comparing two infinite sets

All the linearly independent eigenfunctions of the parity operator $\mathcal{P}$ form an infinite set and all the linearly independent eigenfunctions of the unit operator $\bf 1$ also form an ...
3
votes
2answers
188 views

Matrix elements of linear operators - orthonormal basis required?

In an early linear algebra class of mine, I learnt that a linear map $\mathcal{A}$ acting on a vector space could be represented by a matrix $A_{ij}$ according to the rule: $$\mathcal{A}({e_j}) = ...