# Questions tagged [hilbert-space]

This tag is for questions relating to Hilbert Space, a vector space equipped with an inner product, an operation that allows defining lengths and angles. It arises naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces having the property that it is complete or closed. Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.

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64 views

### Confusion about the Wigner-Eckart theorem

Background This will be a lengthy thread, but I made sure that all 3 questions are related to each other and related to the same topic. I currently encountered the W.E-theorem and while I do ...
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### Whats the meaning of the 1 Ket? [closed]

I am talking this one: $|1\rangle$. If I have 2 orthonormal states $|1\rangle$ and $|2\rangle$ in the 2D Hilbert space, does that imply the vector $\vec{\psi_n}=(1,2)$, if I would like to solve the ...
379 views

### Question about the kinetic energy operator

The Kinetic Energy Operator is essentially self-adjoint. Under what circumstances does it have a unique extension? 51 views
+50

### String theory hilbert space - Gas of free gravitons

I am trying to understand the arguments given in MAGOO in chapter 3.4.1(Hilbert Space of String Theory). The authors give descriptions of the Hilbert space of String Theory when we consider our theory ...
65 views

### What is the role of Hermitian Hamiltonians in relativistic QFT?

In single-particle quantum mechanics, the probability of finding the particle in all space is conserved due to the hermiticity of the Hamiltonians (and remains equal to unity for all times, if ...
1 vote
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### Hilbert space of a diatomic molecule

In molecular quantum mechanics, it is very common to model a diatomic molecule as a two-level harmonic oscillator with vibrational levels lying within the electronic states: In most of the textbooks ...
271 views

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### Why sum of squares of the magnitudes of Fourier coefficients in Infinite Square Well equals one but it is not so in regular Fourier analysis?

My question is basically this.. In regular math, Fourier Coefficients give the "amount" a particular frequency is available in any periodic signal. The squares of sum of coefficients is not ...
1 vote
53 views

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### Spherical harmonics integral

I've been struggling with this integral $$\int_0^{2\pi}\int_0^{\pi} \sin\theta~ e^{-i\phi} Y_{l m}(\theta,\phi) Y^*_{l'm'}(\theta,\phi) ~d\theta ~d\phi$$ I've tried to use the definition of the ...
1 vote
34 views

### Degeneracy of wavefunction in 1 dimension

Suppose we have a one-dimensional bound state, with the degenerate eigenstates given by $\psi(x)$ and $\phi(x)$. Using the Wronskian, we can show that there is no degeneracy, as the two functions are ...
1 vote
39 views

### Definition of a wave packet

In Shankar's QM book page 168, the author stated a wave packet is any wave function with reasonably well-defined position and momentum. What does he mean by resonably well-defined position and ...
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1 vote
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### State time-evolution in the Interaction picture

What is the Schrödinger-like equation $$i\frac{d}{dt}|\psi(t)\rangle_I=V_I|\psi(t)\rangle_I$$ telling us for the behavior of the interaction picture state vectors, $|\psi(t)\rangle_I$, at infinity/...
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### How would I normalize this ket vector? [closed]

So I am given the vector: $$|Ψa⟩ = |x⟩ + |y⟩ − |z⟩$$ And I need to normalize it. I know that I have to take the dot product of the vector with itself (and it needs to equal 1) but how would I do this ...
1 vote
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### Common eigenstate of incompatible observables

In many resources I have seen that incompatible observables cannot have a common eigenbasis set, but may share one or few eigen states. I followed the thread Can incompatible observables share an ...
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### Are qubits just analog, continuous classical bits?

Topologically, classical bits (cbits) are essentially special cases of qubits restricted to the poles of the Bloch sphere. However, this restriction doesn't seem to be classical per se, but is simply ...
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### Differential equations for spontaneous emission for a three-level system

what is the differential equation describing the spontaneous emission of a quantum mechanical three-level system ? Let $|C_i|^2$ ($i=1,2,3$) be the probabilities of the atom being in state $i$. My ...
44 views

### $y$ Pauli Operators Eigenvectors - How are they orthogonal?

I am struggling to obtain that the eigenvectors of the Pauli $y$ operator are orthogonal, and would appreciate guidance on where I am going wrong. I have calculated the eigenvalues as: 1, -1 And ...
157 views

### What is the Hilbert space in quantum field theory?

My understanding is that in classical field theory, we study a classical field $\phi(x,t)$ where for each $x\in\mathbb{R}^3$, $t\in\mathbb{R}$, $\phi(x,t)$ is a scalar. In quantum field theory, we ...
73 views

### Recommendations for Algebraic quantum mechanics book

I am familiar with quantum mechanics and quantum information at the level of Sakurai and Preskill's lecture notes / Nielsen and Chuang. I want to study the $C^*$ algebraic formulation of quantum ...
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### Inner product evaluation in QM

On wikipedia on the page for inner product it states that for any two $x,y$ in a vector space $V$ the inner product $(\cdot , \cdot)$ satisfies $(ax, y) = a(x,y)$ where $a\in\mathbb{C}$. The inner ...
Suppose I have two situations: one where two qubits, $q_A$ and $q_B$, exist independently (on separate sides of the quantum chip, maybe), and one where they exist with some coupling between them. And ...