Questions tagged [hilbert-space]

Question 1

if you write the recurrence as a homogeneous linear system $A\mathbf x=0$, a necessary and sufficient condition for proportionality of (nonzero) solutions is that $\dim\ker A=1$. For a simple ...

Question 2

Let's say we have some non-Hermitian Hamiltonian $H(\vec{\xi})$ depending on a set of parameters $\vec{\xi}$. Whenever $H(\vec{\xi})$ is not diagonalizable (two or more eigenstates coalesce), let's ...

Question 3

If we represent spin quantum state of a particle in $\pm z$ direction with $\vert\pm\rangle$ then we know that the state vectors in remaining $x$,$y$ directions would be such that: $$\vert\langle S_x;...

Question 4

I'm trying to solidify my understanding of the path integral formalism in Quantum Field Theory, and I've run into a conceptual paradox regarding the definition of initial states. I would be grateful ...

Question 5

From wikipedia https://en.wikipedia.org/wiki/Wigner%27s_theorem For unitary case $$\langle U \Psi, U \Phi \rangle = \langle \Psi, \Phi \rangle .\tag{1} $$ If I apply the definition of adjoint https://...

Question 6

If a mode function of the light is given by $\psi_{\mathbf k}(x^\mu)=ce^{ik_\mu x^\mu}$, where the degrees of freedom of polarization are suppressed, it can be normalized by requiring $\left <\psi_{...

Question 7

Suppose we compute an expectation value of $r_{12} r_{13}^{-1}$ over a wave function $\phi_p (1) \otimes \phi_q(2) \otimes \phi_r (3)$, we denote it as $$\langle pqr | r_{12} r_{13}^{-1} |pqr \rangle. ...

Question 8

If I have a quantum system and trace out some degrees of freedom (i.e. in the Green function formalism), I get a subsystem governed by an effective Hamiltonian. Now, if we assume that the full system ...

Question 9

I am a Math student, new at Quantum Mechanics, and I am having some troubles understanding the physical meaning of the notion of “complete set of compatible observables". I know its mathematical ...

Question 10

I am learning QM and trying to understand the (absence of) SSB in QM. I have read statements from several posts in the forum about the absence of SSB in few-body QM (and I'm not sure if they are ...

Question 11

This is from page 127 of Sakurai QM: How did they obtain this result? I understand the factor comes from the Hamilton on the LHS, but how were they able to pull out the factor out of the bra-ket like ...

Question 12

Using Peskin+Schroeder as a reference. Bear with me, there may be multiple mistakes in my discussion. But the underlying question should be clear - it's really just the title. By analyzing the ...

Question 13

A common way to derive Heisenberg's EoM is to start from the operator in Heisenberg picture: $$ A_H(t)=U^\dagger(t)A_SU(t).\tag{1} $$ We can get: $$ \frac{dA_H}{dt}=i[H ,A_H] .\tag{2} $$ Can we derive ...

Question 14

We know that the inner product of a basis vector of an observable or operator with itself should be 1 and should be 0 when inner producted with any other basis vector of the same observable is $0$.But ...

Question 15

Unitary transformations conserve the inner product structure of a set of vectors, they only change the direction of the vectors, i.e. rotate them all in the same way. A unitary transformation $U$ can ...

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Questions tagged [hilbert-space]

Problem in understanding the proof of Wigner-Eckart in Sakurai

Quick question about self-orthogonality of non-Hermitian Hamiltonians

What would Spin Operator and Spin State Vectors be in higher dimensions?

QFT Path Integrals: How to define a precise initial state if vacuum fluctuations can alter it?

Does Wigner's theorem only imply left inverse?

Is it valid to keep $k_1$ and $k_2$ when considering a light propagating along $z$ axis?

A paradox in using completeness relation $\sum |\rangle\langle|=1$ of quantum mechanics

Subsystem in ground state?

Physical meaning of a complete set of compatible observables

Spontaneous symmetry breaking in quantum mechanics

Question about Schrödinger equation with scalar and vector potentials from Sakurai

What are the units of a state in QFT?

How to derive Heisenberg's EoM from canonical equation? [closed]

Magnitude of basisvectors [duplicate]

Vector transformations that preserve norms but changes inner products between different vectors

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