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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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Eigenvalues of the Hamiltonian

Is every eigenvalue of the Hamiltonian a form of energy? If not are there values of the Hamiltonian that do not correspond to the energy of the system?
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Diagonalizinng a matrix [closed]

Given a matrix $A=\begin{pmatrix} a_{11}& a_{12}\\ a_{21} & a_{22} \end{pmatrix}$ with eigenvalues $\lambda_1$ and $\lambda_2$. If we want to diagonalize this matrix, the standard procedure ...
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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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What is energy in quantum mechanics?

Is it wrong to say energy is the expectation value of Hamiltonian? Or should I say energy is the eigenvalue of Hamiltonian?
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Arriving at the Quantum Mechanial Potential From The Energy Eigenvalues [duplicate]

In Quantum Mechanics, we know that given a potential we can solve the eigen value problem to find out the energy eigen values and eigen functions. Now suppose in an experiment we have information only ...
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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^{...
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Creation operator acting on a coherent state. Occupation number operator

For a coherent state $$|\alpha\rangle=e^{-\frac{|\alpha|^{2}}{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}(a^{\dagger})^n}{n!}|0\rangle$$ I want to find a simplified expression for $a^{\dagger}|\alpha\...
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Some questions on coherent states and corresponding Hilbert spaces. Reproducing kernal

I have a few questions related to coherent states. I use this source https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_5.pdf. Using standart inner product $\langle\cdot|\cdot\rangle$ ...
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Confusion of measuring two quantities on a quantum system

Let's say there are two observables corresponding to two operators A and B, and let's say my system is in a state Phi where with probability 1 if I measure A I get 3 (let's say 3 Joules), If I ...
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1d Ising model: Transfer matrices

we came across a peculiarity when calculating the partition function of $N$ spins $s_i=\pm1$ with Hamiltonian $$H=-J\sum_{i=1}^Ns_is_{i+1}$$ where we impose periodic boundary conditions such that $s_{...
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Eigenvalues of the thermal state density operator

We define the thermal density operator as $$\tau(\beta) = \frac{e^{-\beta H}}{\mathrm{Tr}(e^{-\beta H})}$$ where $H$ is the systems Hamiltonian. Today I was told that the eigenvalues of the ...
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What is the term for a particle spin's uncollapsed position? What is the orbiting “thing”?

I'm not sure if I have the correct visual model, but I imagine that a particle spin can be represented by a single point on the orbit, or by a superposition state (like a random plane through a corner ...
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Applying Sylvester's theorem in quantum mechanics

A $2d$ system consists of $N$ identical cells arranged linearly in series. The transfer matrix of a single cell is an unitary Hermitian $2$x$2$ matrix with eigenvalues $\exp(±i\theta)$. I need to use ...
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Quantum Potential - how to find its eigenstates?

I am studying the Pöschl-Teller potential in spherical coordinates and doing a change of variable is enough to put it in another sort of differential equation and thus, obtain the solution. The point ...
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2answers
104 views

Conceptual understanding of Schrödinger equation

So I followed this lecture: https://www.youtube.com/watch?v=qu-jyrwW6hw which starts of with the statement: If you have a Schrödinger equation for an energy eigenstate you have $$-\frac{\hbar}{2m}...
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Does the eigenbasis associated with an observable changes after measuring a different observable?

Suppose a system is initially in a superposition: $$\psi(x) = \sum\limits_{i}|c_i\phi_i(x)\rangle$$ After a position measurement, the wave function collapses to one of the position eigenfunctions,$\...
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Expansion of the infinite square well [closed]

I was studying the expectation value of the energy of a particle in the groud state of the infinite square well after its expansion in terms of width (from $a$ to $2a$), which is: $$\langle H\rangle= ...
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Why is this function an eigenfunction of $\hat{L}_{z}$?

$$\Psi(\varphi)=\frac{1}{\sqrt{2\pi}}(\sin\varphi-\cos\varphi)$$ I am not able to see why the above function is an eigenfunction of $\hat{L}_{z}$ and which is its eigenvalue. I've been trying with ...
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Applications of Hamiltonians proportional to nth order momentum

Has anyone come across any physical applications of a linear theory where Hamiltonians are proportional to an arbitrary order of momentum? This paper https://michaelberryphysics.files.wordpress.com/...
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What exactly are control functions (used for parametrization)?

Let us consider a system in state $\rho$ with an internal hamiltonian $H_0$ on which we apply a cyclic, unitary evolution $H_t = H_0 + V(t)$ Where $V(t)$ is a time dependent external potential for ...
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If $L_z$ has a $0$ eigenfunction, since $[L_x, L_y] = i\hbar L_z,$, then can $L_x, L_y$ have a simultaneous eigenfunctions?

In the lecture Quantum Mechanics by Dr. Adams in ocw.mit.edu, in the 16th lecture at 7:11, it is stated that since $$[L_x, L_y] = i\hbar L_z,$$ there is no state s.t it is eigenfunction of both $L_x, ...
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Woods-Saxon energy levels

I'm trying to calculate the bound state energy levels for a Woods-Saxon potential, given by: $$V(r) = -\frac{V_{0}}{1+\exp(\frac{r-R}{a})}$$ I'm using Numerov's matrix method, which led me to: $$H\...
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Can a general many-body Hamiltonian with quadratic and biquadratic terms be diagonalized?

Can an arbitrary many-body hamiltonian in second quantization form with quadratic and biquadratic terms $$H=\sum_{v_1,v_2} \alpha_{v_1 v_2}\ c_{v_1}^{\dagger}c_{v_2}+ \sum_{v_1,v_2,v_3,v_4}\beta_{v_1 ...
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Wigner proof of the non-existence of finite unitary representation of the Lorentz group

I am reading Wigner's paper ”On unitary representations of the inhomogenous Lorentz group” (Annals of Mathematics, Vol. 40, No.1, p. 149) found here: https://www.maths.ed.ac.uk/~jmf/Teaching/Projects/...
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What do I get by multiplying a 0 operator on a 0 eigenvector?

I don't know how to write the equation form. Assuming my notation as Dirac notation, what do I get from $$ ( 0 | 0 | 0 ) ~?$$
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What is eigenvalue and eigenfunction in quantum mechanics?

What is the use of eigenvalue and eigenfunction in quantum mechanics specially Schrodinger equation? What is the physical meaning of having an eigenvalue and eigenfunction in Schrodinger equation?
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36 views

Energy of Free-electron Gas - Landau Levels in 3D

so i am looking into Landau Diamagnetism and am reading Dupre's paper. I am slightly confused at where he has got a term in his value of E from. He states that: $$ E=(n+1/2)\hbar\omega+\hbar^2k_z^...
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50 views

How to calculate the ground state of Ising model at non-zero temperature

I'm studying the quantum Ising model, i.e. with Hamiltonian $H= -h\sum_{i}X_i-\sum_{\langle i,j\rangle}Z_iZ_j$. I know conceptually how to compute the ground state of the Ising model at zero ...
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1answer
114 views

Diagonalization of a matrix with operators as elements

How to diagonalize a hamiltonian matrix that has differential operators as elements? My matrix looks something like: \begin{bmatrix} A \frac{d^{2}}{{d\theta}^{2}}+ B_{1} & a\cos{(b\theta +c)}\\ a\...
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1answer
86 views

Eigenvalues and functions in quantum mechanics [closed]

How do I determine if $\psi(x)$ is a eigenfunction of some operator and find the corresponding eigenvalues, where $\psi(x)$ is the wave function of free particle (potential = zero).
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1answer
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Finding measurements in non-Hermitian operators

I know how the measurement postulate in quantum mechanics works, in regard to hermitian operators, but what if an operator is non-hermitian? Can i apply the following reasoning? If an operator is ...
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1answer
91 views

Symmetry-breaking matrix/operator deformations: uniquely splitting eigenspaces into smaller ones?

Operators used in quantum mechanics, like Hamiltonian or angular momentum operator, usually have huge degeneracy of eigenspaces (symmetry inside them) - bringing a question of possibility to uniquely ...
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1answer
38 views

Is the ground state energy always larger for the system with higher potential energy?

Say we have two Hamiltonians $\hat{H}_1$ and $\hat{H}_2$ that differ only in their potential energies and $$V_2(x) > V_1(x)$$ for all $x$. Is the energy of the ground state of system 2 necessarily ...
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1answer
85 views

Eigenstates of a Hamiltonian [closed]

For a particle with a spin of 1/2, which was exposed to both magnetic fields $B_{0}=B_{z}e_z$ and $B_1=B_xe_x$ I already found the eigenvalues of its Hamiltonian which is given by \begin{...
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Am I correct to say that ladder operators have complex eigenvalues?

From the definition: $$\left. \begin{array} { l } { \hat { L } _ { + } = \hat { L } _ { x } + i \hat { L } _ { y } } \\ { \hat { L } _ { - } = \hat { L } _ { x } - i \hat { L } _ { y } } \end{array} \...
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1answer
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How to diagonalise a hamiltonian which posesses symmetry?

I have a large hamiltonian but I know that it posseses some symmetries. How do you reduce the hamiltonian in order to find the eigenenergies?
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mutual coherence

I read Vertex-Frequency Analysis on Graphs (arxiv link) Shuman, David I, et al. “Vertex-Frequency Analysis on Graphs.” Applied and Computational Harmonic Analysis, vol. 40, no. 2, 2016, pp. 260–291....
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Quantum Spherical Pendulum [closed]

I have trouble with finding the eigenstates of a spherical pendulum (length $l$, mass $m$) under the small angle approximation. My intuition is that the final result should be some sort of ...
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2answers
355 views

Eigenvalues of momentum operator

I had a homework problem in my intro QM class, basically asking me to find which of a given set of functions were eigenfunctions of the momentum operator, $\hat{p_x}$. For example, $$ \psi_1 = Ae^{ik(...
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1answer
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Significance of eigenvalues of an observable Of a wavefunction [closed]

What really is meant by eigenvalue of an observable? Does it mean that everytime we measure a value of an observable the result obtained is equal to the eigenvalue of the observable?
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Why the eigenvalues of hamiltonian are discrete in bounded systems (only discrete energy levels)? [duplicate]

In one dimensional motion the general potential is given as in the figure above When energy is between V min and V 1 the energy levels are discrete like in the potential barrier and well example or ...
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1answer
104 views

Reconcile a pair of two-qubit boundary-state separability probability analyses

It is now clearly well-established--though formalized proofs are still largely lacking—that the probability, with respect to Hilbert-Schmidt measure, that a generic two-qubit state is separable/...
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1answer
118 views

Under what conditions is a wavefunction $\psi(x)$ equal to the probability amplitudes $a(x)$?

For context, consider a general expansion of a wavefunction into continuous eigenstates of position, $\phi(x_m,x)$, multiplied by continuous probability amplitudes, $a(x_m)$ $$\begin{align}\psi(x) &...
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1answer
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A naive question on the eigenvalues of fermionic operators?

Let $A$ be a fermionic operator which is a product of odd number of fermion operators or a summation of them, say $A=C_{i_1}^{\dagger}\cdot \cdot\cdot C_{i_m}^{\dagger}C_{j_1}\cdot \cdot\cdot C_{j_n}...
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2answers
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Something special about energy eigenstates when it comes to time evolution?

A particle is subject to an infinite square well potential with $$V(x)= \begin{cases} 0 & −a \lt x \lt a\\ \infty & \,\,\,\,\text{otherwise} \end{cases}$$ At a time $t=0$ its ...
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2answers
94 views

On the spectrum of a quantum mechanical system [duplicate]

Can the spectrum of a quantum mechanical operator contain both real and complex numbers?
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2answers
181 views

Proving orthogonality of eigenstates of a Hamiltonian

Suppose we have $\Psi_{1}$ and $\Psi_{2}$ which are eigenstates of some (self-adjoint) Hamiltonian $\hat{H}$ with unequal eigenvalues. Could you explain me how can I prove that these arbitrary $\Psi_{...
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Does the trace distance specify a unique state

In quantum information, we frequently use the trace distance (see definition) to look at how similar two states are. If I had a known complete set of states $\{\rho_i\}$ and some unknown state $\...
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82 views

How is every finite linear combination of eigenstates of H a bound state?

This is all i could find in my lecture notes about this. Its not very useful tbh. How can i show this? Thanks
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3answers
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Eigenvalue problem in Hilbert Space

In page 71 of Shankar's Principles of Quantum Mechanics, the author states the following(kindly take a look at page 70 because the following is a part of an example problem): The allowed ...