# Tagged Questions

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, ...

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### Eigenvalues of Hermitian operators are real and the dependence/independence of boundary conditions

Without reproducing proofs: Eigenvalues of a Hermitian operator are real (proof does not rely on the boundary conditions). The momentum operator is Hermitian (proof does not rely on the boundary ...
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### Oscillator in Energy Basis Lowering and Raising Operators

On page 205 of Shankar's Intro to Quantum Mechanics, equation 7.4.12 does not make sense to me. I understand why a|e> is an eigenvector and why e-1 is its Eigenvalue, but I don't understand how that ...
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### Eigenvalue equation for kinetic and potential energy

In Boas' Mathematical Methods there is a section on linear algebra in which it is stated that we can write the eigenvalue equation for a set of springs using the kinetic energy and the potential ...
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### In quantum mechanics how can eigenfunction, eigenvalues, matrix methods give us values of real physical quantities? [closed]

Eigenfunction, eigenvalues, eigenstates & matrix methods used in quantum mechanics seems purely mathematical.How can they give us values of real physical quantities in quantum mechanics?
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### Neutron in a magnetic field (Schrödinger Equation, Eigenstates, Eigenvalues).

Consider the spin of a neutron in a magnetic field $\vec{B}$. A neutron is a neutral particle with the mass of a proton and the spin $\frac{1}{2}$. The Hamiltonian is $H=\mu_n\vec{S}\cdot\vec{B}$...
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### Eigenvalues of spherical harmonics in $d$ dimensions

I'm working on the Schrodinger equation for a hydrogen atom in a $d$-dimensional space, so I'm interested in the possible eigenvalues of the angular momentum part of the $d$-dimensional Laplace ...
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### Zeeman effect - eigenstates and its degeneracy - with and without magnetic field

Consider a hydrogen atom in a homogeneous magnetic field $\vec{B}=B\vec{e_z}$. Using the coulomb gauge ($\nabla \vec{A}=0$) we can take $\vec{A}=\frac{1}{2}\vec{B}\times \vec{r}$ as a vector potential....
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### Random operator in heisenberg/schrödinger picture (heisenberg's equation of motion) [closed]

Consider a system whose hamiltonian isn't explicitly dependent on time. Let A be the operator for the eigenvalue a in the Schrödinger picture and $A_H=U^\dagger A U$ the corresponding operator in the ...
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### How to understand Preskill's argument for degeneration of eigenstates?

In his notes on topological quantum computation on page 18, Preskill uses the "commutator" $T_2^{-1}T_1^{-1}T_2T_1 = e^{-2 i \vartheta}$ to show that the eigenstates of $T_1$ are degenerate. But I don'...
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### Reasoning behind taking the Fourier transform of the fermionic operators for a circular $1$D spin chain [closed]

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring. To compute the complexity of the algorithm ...
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