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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46
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7answers
3k views

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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1answer
2k views

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 ...
11
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2answers
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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 ...
18
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1answer
2k views

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 ...
17
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4answers
12k views

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') ...
33
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7answers
4k views

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 ...
18
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4answers
10k views

How to tackle 'dot' product for spin matrices

I read a textbook today on quantum mechanics regarding the Pauli spin matrices for two particles, it gives the Hamiltonian as $$ H = \alpha[\sigma_z^1 + \sigma_z^2] + \gamma\vec{\sigma}^1\cdot\vec{\...
15
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3answers
2k views

Eigenstate of field operator in QFT

Why don't people discuss the eigenstate of the field operator? For example, the real scalar field the field operator is Hermitian, so its eigenstate is an observable quantity.
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3answers
6k views

Momentum of particle in a box

Take a unit box, the energy eigenfunctions are $\sin(n\pi x)$ (ignoring normalization constant) inside the box and 0 outside. I have read that there is no momentum operator for a particle in a box, ...
6
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1answer
2k views

Can any physical rigid body be represented by an ellipsoid with the same angular dynamics?

According to wikipedia, the inertia tensor of an ellipsoid with semi-axes $a,b,c$ and mass $m$ is $$\left[\begin{array}{ccc} \frac{m}{5}(b^2+c^2)&0&0\\ 0&\frac{m}{5}(a^2+c^2)&0\\ 0&...
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1answer
5k views

Is there a simple way of finding the eigenstates of the creation and annihilation operator in QM?

How can I find the eigenstates of creation and annihilation operator in QM? My attempt: Such eigenstate will obey: $$ a^{\dagger} |\psi \rangle = \alpha |\psi \rangle. $$ We can expand $|\psi \...
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1answer
798 views

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 ...
13
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4answers
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Why are the inner products of the eigenfunctions of an operator with a discrete eigenvalue spectrum guaranteed to exist?

I was reading through a textbook, and the statement was made that the inner products are guaranteed to exist if the eigenvalue spectrum of the operator is discrete. I have come across no support for ...
14
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3answers
798 views

How do we know that we have captured the entire spectrum of the Harmonic Oscillator by using ladder operators?

Consider standard quantum harmonic oscillator, $H = \frac{1}{2m}P^2 + \frac{1}{2}m\omega^2Q^2$. We can solve this problem by defining the ladder operators $a$ and $a^{\dagger}$. One can show that ...
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2answers
13k views

Numerical solution to Schrödinger equation - eigenvalues

This is my first question on here. I'm trying to numerically solve the Schrödinger equation for the Woods-Saxon Potential and find the energy eigenvalues and eigenfunctions but I am confused about how ...
9
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4answers
970 views

Eigenvalues of the Lagrangian?

It is often stated that the Lagrangian formalism and the Hamiltonian formalism are equivalent. We often hear people talk about eigenvalues of Hamiltonians but I have never ever heard a word about ...
8
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2answers
344 views

How can an inverted anharmonic potential $V(x)=-x^4$ have discrete bound states?

I've been watching the lectures on mathematical physics by Carl Bender on youtube where he uses the non-Hermitian Hamiltonian methods to prove that the inverted anharmonic potential $V(x)=-x^4$ has a ...
5
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1answer
323 views

Quantum mechanics - measuring position

I am watching Susskind's Stanford Lectures on quantum mechanics. The eigenvectors (eigenfunctions) of the position operator are of the form $\delta(x-k)$. But $$\int\delta^{*}(x-k)\delta(x-k)\, \...
4
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2answers
554 views

Hamiltionian spectrum in unstable systems

I have heard that the eigenvalue of Hamiltonian in an unstable system can contain an imaginary part corresponding the tunneling. Is that true? If it is the case, then I am very confused about it. Let ...
4
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0answers
151 views

What's the space of eigenvalues/field configurations for a fermion?

In the Schrödinger picture of quantum field theory, the field eigenstates of a real scalar field $\hat\phi(\mathbf x)$ with $\mathbf x \in\mathbb R^3$ are the states $\hat\phi(\mathbf x)|\Phi\rangle=\...
3
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4answers
452 views

Help understanding proof in simultaneous diagonalization

The proof is from Principles of Quantum Mechanics by Shankar. The theorem is: If $\Omega$ and $\Lambda$ are two commuting Hermitian operators, there exists (at least) a basis of common eigenvectors ...
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2answers
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Eigenvalues of a quantum field?

Fields in classical mechanics are observables. For example, I can measure the value of the electric field at some (x,t). In quantum field theory, the classical field is promoted to an operator-valued ...
8
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1answer
521 views

Role of physics in the zeta function $\zeta$ and the Riemann hypothesis

Hilbert and Polya suggested a physical way to verify the Riemann hypotesis about $\zeta(x)$. If the Riemann hypotesis is true, we can state all eigenvalues of physical problems are real. What is the ...
8
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1answer
538 views

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 ...
3
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1answer
1k views

WKB approximation for multiple turning points

I'm working on a numerical program which approximates the eigenvalues of a Schrödinger equation by making use of the WKB approximation formulas. For example, if the Schrödinger equation is $$ y''(x) = ...
9
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1answer
574 views

Why are the eigenvalues of a linearized RG transformation real?

The RG transformation $R_\ell$ maps a set of coupling constants $[K]$ of a model Hamiltonian to a new set of coupling constants $[K']=R_\ell[K]$ of a coarse-grained model where the length scale is ...
7
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1answer
202 views

Can I determine the potential term in the Schrödinger equation based on the eigenvalues? [duplicate]

Let's imagine I knew a certain system could be described by a one-dimensional Schroedinger equation. I know the mass/momentum term, but not the shape of the potential. Further for some reason I know ...
3
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2answers
2k views

Non-Degeneracy of Eigenvalues of Number Operator for Simple Harmonic Oscillator [duplicate]

Possible Duplicate: Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate? I'm trying to convince myself that the eigenvalues $n$ of the number operator $N=a^{\dagger}a$ for ...
3
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2answers
206 views

Are there any non-physical eigenstates of interacting QFT?

While reading about the Källén-Lehmann representation I came across the definition of eigenstates in general QFT. As $\vec{p}$ (total momentum) and $H$ commute they can be simultaneously diagonalized, ...
2
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3answers
967 views

Spin in magnetic field and eigenvalues

We have some arbitrary quantum state, lets say $$\vert\Psi\rangle=\alpha_{1}\vert\uparrow\rangle+\alpha_{2}\vert\downarrow\rangle= \begin{pmatrix} \alpha_{1} \\ \alpha_{2} \\ \end{pmatrix}$$. And ...
2
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1answer
52 views

What is the form of the $n$-th order term of the perturbation series of an eigenvalue?

Suppose I have a matrix given by a sum $A=D+\epsilon B$, where $D$ is diagonal and $\epsilon$ is small, and I want the eigenvalues of $A$ as power series in $\epsilon$. The leading order is just the ...
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2answers
601 views

What is the meaning of pre-tension for a stiff membrane?

On one hand, I know that the tighter a drum head is stretched, the higher its natural frequencies. This relation is given by: $$f_{ij}=\frac{k_{ij}}{2\pi R}\sqrt{\frac{T_0}{h\rho}}$$ where $k_{ij}$ ...
5
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2answers
3k views

Quantum Mechanics Notation for BRA KET

I've been given this homework problem, but I do not understand its notation. Please perform the following where the wavefunctions are the normalized eigenfunctions of the harmonic oscillator ...
4
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1answer
636 views

What is the problem of having an inertia tensor not satisfying the triangle inequality?

It is well known that rigid body inertia tensors are 3 by 3 positive semidefinite matrices, which is the same as saying that their eigenvalues are all non-negative. A little less known is the fact ...
4
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3answers
7k views

Finite potential well problem - calculating the ground state

1. The problem statement, all variables and given/known data Electron of is in a 1-D potential well of depth $20eV$ width $d=0.2 nm$ in his ground state $N=1$. What is the energy of the ground ...
3
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0answers
49 views

Chua's Circuit: an inequality ensuring that the equilibrium is not stable

According to Kennedy's Robust op-amp realization of Chua's circuit(1992), the differential equations satisfied by several physical quantities in Chua's circuit are $$\begin{aligned} C_{1} \frac{d v_{...
2
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2answers
1k views

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 ...
2
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3answers
3k views

Why do electrons occupy in discrete energy states?

Why can't there be any continuous energy band in an atom?
2
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1answer
150 views

Why does the first radial excitation of a particle in a 2D annulus $a<r<b$, for $b\gg a$ lie between the second and third azimuthal excitations?

Consider the quantum mechanics of a massive particle confined by infinite potential walls to a 2D annulus $a<r<b$, for which the Hamiltonian's eigenfunctions obey the stationary Schrödinger ...
2
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1answer
75 views

Optimizing the second, third,… eigenvalues - applications

I'm working on some topics related to spectral optimization as a function of the domain. For example it is known for almost a century (lord Rayleigh and Faber, Krahn) that the shape which minimizes ...
1
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1answer
397 views

Single particle operator in second quantization

I want to understand why we write in the formalism of second quantization for a single particle operator \begin{equation} \hat H=\sum_i \varepsilon_i \hat a_i^{\dagger} \hat a_i \end{equation} where ...
0
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2answers
161 views

Addition of angular momenta and Clebsch-Coefficients

If we consider two angular momentum operators $\hat{J}_{1}$ and $\hat{J}_{2}$ and where $J := J_{1} \otimes 1 + 1 \otimes J_{2}$ where respectively we have common eigenstates $|j_1j_2;m_1 m_2 \rangle$ ...
0
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1answer
41 views

How to write “postselection” operator?

Suppose, I wish to know an operator, which eigenvalue is 1 if state is exactly F and 0 ...
6
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3answers
1k views

Imaginary Eigenvalue Of A Hermitian Operator

The eigenfunctions of a Hermitian operator are real. But consider a function $\psi(x)=e^{-\kappa x}$, $x\in\mathbb{R}$, where $\kappa$ is a real constant. Then, $$\hat p \psi(x)=-i\hbar \frac{d}{dx}e^{...
4
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2answers
2k views

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 ...
2
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0answers
171 views

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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1answer
71 views

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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3answers
169 views

Superposition principle forbids quantisation?

Apparently bound states in quantum mechanics require energy states to be discrete. That means energy in such systems is quantized, right? However, say that we have a superposition of energy ...
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0answers
105 views

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^{...
1
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1answer
86 views

How to write QM operator if I know all of it's eigenfunctions?

Suppose I have selected enough orthogonal functions in representation of operator A and I want to derive operator B which has ...