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, and the space is complete. It arises naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces having the property that it is complete. Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.
5,101 questions
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Directionality of operator in a braket
I have this operator $Y_1^1(\theta,\phi)N_-$, where $N_-$ is the lowering operator for $|l,m\rangle$ and $Y_1^1(\theta,\phi)$ acts just as a multiplication. I want to calculate: $\langle 0,0|Y_1^1(\...
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Classical Mechanics in the Mackey's axiomatization of a physical system
G. L. Naber in its Quantum Mechanics gives an axiomatization of a physical system in mathematical terms. The story goes like this: Definition. A physical system is composed of: A collection $ \...
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Meaning of average pressure in statistical mechanics
Kittel and Kroemer derive the pressure of a statistical state in the following way: They assume a volume compression of a system such that the quantum state of the system is maintained at all times; ...
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Confusion regarding expression of Expectation Value in Quantum Mechanics
So, initially when I first learned Quantum Mechanics, the expectation value of the operator corresponding to a physical observable was given by : $$\langle \hat{Q} \rangle = \frac{\int_{-\infty}^{\...
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Beyond Hilbert space: what kind of vector space is needed for Dirac's Transformation Theory? [duplicate]
It is fairly common-place amongst physicists to consider physical observables that may take a continuum of values $x$ and follow Dirac in representing a physical state as an integral over a set of ...
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Lorentz transformation of Hilbert space operators
I want to address a confusion that has arisen in certain questions of mine on this site, which you can find "here" and "here". My confusion is related to the transformation under ...
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Lorentz Transformation of quantum vector field
Consider a vector field $A^\mu(x)$. Its Lorentz transformation is given by \begin{equation}\tag{1} A'^{\mu}(x') = \Lambda^\mu_{\:\nu}\,A^\nu(x), \end{equation} where $x'\equiv x'^\mu = \Lambda^\mu_{\:\...
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Why is $\hat{a}_- \psi = 0$ in a QHO? [duplicate]
My question is as follows: The ladder operators seem to create a new wave function and also new energy states. So when it acts on the wave function of the ground state, what does $\hat{a}_- \psi = 0$ ...
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Incompleteness of positive-energy solutions to Klein-Gordon equation
In non-relativistic Quantum Mechanics, plane waves (didn't attempt to normalized below) or eigenstates of momentum operator form a complete set of basis of the state space $L^2(\mathbb{R}^3)$ (...
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Dividing multi-particle states into a sum of states with particles of definite number
In Peskin & Schroeder's book (7.1 , p212) in the studying of the analytic structure of two-point correlation function, they generalize the completeness relationship of 1 particle states $$ \left( ...
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How to prove $\langle p,s,-|\Psi (x)|0\rangle =v_s (p)e^{-ipx} $?
M. Srednicki in his book "Quantum field theory" on page 265 mentions that for validity of LSZ formula we must have (for a Dirac field): $$\langle p,s,-|\Psi (x)|0\rangle =v_s ({\bf p})e^{-...
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Effect of rotation in Hilbert space
Let's assume I have a quantum state $|\alpha \rangle$. I rotate it using the rotation operator and I get a state $| \alpha \rangle_{R}$. I know that the expectation values of all vector operators in ...
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Existence of a state in the algebraic formulation of quantum theories
In the algebraic formulation of quantum theories, we define observables as self-adjoint elements of a unital $*$-algebra $\mathcal{A}$, and (algebraic) states as positive normalized complex-linear ...
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An intuitive derivation of cross-section formula in quantum field theory [closed]
I've seen rigorous derivations of using amplitude from Feynman diagram to calculate cross section (e.g. Peskin & Schroeder's book), However, none of the derivation is intuitively clear, and I am ...
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How do we express the quantum state of a particle in momentum eigenfunctions basis given that eigenfunctions of momentum are not square-integrable?
As far as I know, for a particle without spin the quantum state of the particle is described by a "point" in the Hilbert space of the equivalence classes of $L^2$ square-integrable functions ...
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