# Questions tagged [eigenvalue]

A linear operator (including a matrix) acting on a non-zero *eigenvector* preserves its direction but, in general, scales its magnitude by a scalar quantity *λ* called the *eigenvalue* or characteristic value associated with that eigenvector. Even though it is normally used for linear operators, it may also extend to nonlinear operations, such as Schroeder functional composition, which evoke linear operations.

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### In quantum mechanics, given certain energy spectrum can one generate the corresponding potential?

A typical problem in quantum mechanics is to calculate the spectrum that corresponds to a given potential. Is there a one to one correspondence between the potential and its spectrum? If the answer ...
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### Schrödinger wavefunctional quantum-field eigenstates

The reason that I have this problem is that I'm trying to solve problem 14.4 of Schwartz's QFT book, which've confused me for a long time. The problem is to construct the eigenstates of a quantum ...
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### Why are eigenfunctions which correspond to discrete/continuous eigenvalue spectra guaranteed to be normalizable/non-normalizable?

These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by ...
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### Discreteness of set of energy eigenvalues

Given some potential $V$, we have the eigenvalue problem $$-\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi$$ with the boundary condition $$\lim_{|x|\rightarrow \infty} \psi(x) = 0$$ If we ...
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### Why do we use Hermitian operators in QM?

Position, momentum, energy and other observables yield real-valued measurements. The Hilbert-space formalism accounts for this physical fact by associating observables with Hermitian ('self-adjoint') ...
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### Why are Only Real Things Measurable?

Why can't we measure imaginary numbers? I mean, we can take the projection of a complex wave to be the "viewable" part, so why are imaginary numbers given this immeasurable descriptor? Namely with ...
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### Quick question on sketching wavefunction in well

Usually for an infinite well, the sketch for n=3 level is this: Now I think if one side of the potential barrier is higher, the particle will be more likely to spend time on the left side than the ...
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### Adiabatic quantum evolution of single photon or biphoton system

The prerequisite for adiabatic quantum evolution of single photon or biphoton system is as follows. We have to prepare a single photon or biphoton quantum system which has a ground and a higher level ...
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### Use of operators in a time-dependent Hamiltonian quantum system

I am given the following Hamiltonian, $$H=H_1=\frac{p^2}{2m}+\frac{1}{2}m\omega_1^2x^2$$ for $t<0$ and $$H=H_2=\frac{p^2}{2m}+\frac{1}{2}m\omega_2^2x^2$$ for $t\geq0$. For some time $t_1(<0)$, ...
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### Numerical exact diagonalization of tight binding Hamiltonian

I want to exactly diagonalize the following Hamiltonian for $10$ number of sites and $4$ number of spinless fermions H = -t\sum_i^{L-1} \big[c_i^\dagger c_{i+1} - c_i c_{i+1}^\dagger\big] + V\sum_i^{...