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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, ...

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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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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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35 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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Eigenvalues of a non-square jacobian matrix that describes the dynamical system

I've been trying to describe a system which has 2 conjugate momentas and 2 positions. By reducing them into 2 dimensions, I was trying to construct the jacobian matrix for two of them independently ...
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1answer
94 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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72 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
76 views

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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86 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
37 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
62 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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139 views

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
55 views

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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41 views

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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98 views

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
185 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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64 views

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
102 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
113 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
80 views

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
174 views

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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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
114 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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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
49 views

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 ...
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1answer
63 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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1answer
100 views

Definition of state of a quantum system

In QM, we solve for the eigen kets of the Hamiltonian operator $\hat{H}$ and say that the state of my system lies in a linear superposition of these eigenstates $\{|n\rangle\}$ as the relation implies ...
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1answer
76 views

Reasoning behind the solution to this Hamiltonian

I am confused about Griffith's solution (Example 10.1 pg 374) for the Hamiltonian: $$H = \dfrac{\hbar\omega_1}{2}\begin{bmatrix} \cos\alpha &e^{-i\omega t}\sin\alpha \\ e^{i\omega t}\sin\alpha &...
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35 views

Are the coefficient in a linear combination of eigenstates uniquely determined?

Good evening, Are the coefficient in a linear combination of eigenstates uniquely determined ? Because, for all the exemples I've already seen ( only in 1D ) like the HO, we only determined the ...
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1answer
46 views

Show that normable eigenstates are not degenerated in 1D [duplicate]

I'm given a particle in a piecewise continous potential $V$ and am supposed to show that normable eigenstates of the Hamilton Operator are not degenerated. My approach was to find two eigenfunctions $\...
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1answer
103 views

Hydrogen atom- Eigenvalue/function relation

I have been given the following Question: The energy eigenstates of the atomic electron are usually described by wave functions $ψ_{nℓm}(r)$. Relate each of $n, ℓ,$ and $m$ to the eigenvalue of a ...
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2answers
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Functions of operators

Consider the function $e^{\Omega}$ where $\Omega$ is a Hermitian operator. We can show that this function is well defined by going to the eigenbasis of $\Omega$ and studying the convergence of the ...
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1answer
28 views

Explicitly prepare a particle in a state other than the eigenstate

Is it possible to explicitly prepare a single system in a state other than the eigenstate? For example, to prepare a particle in such a way that it will show spin up 80% of the time, and down 20% of ...
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1answer
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Identify a $|Ψ(0)⟩$ with $A|Ψ(0)⟩≠a|Ψ(0)⟩$ $\forall$ $A$ & $|Ψ(0)⟩=\sum a_j\lvert\chi_j\rangle$ for some $A$ and its eigenstates$\lvert\chi_j\rangle$

Is it possible to put a quantum system in a state at time $t=0$, which is not the eigenstate of any observable, but at the same time can be linearly expanded using the eigenstates of some observable? ...
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3answers
252 views

Hermitian operator in an orthonormal eigenbasis

In page 36 of Shankar's Principles of Quantum Mechanics is given a theorem: Theorem 10. To every Hermitian Operator $\Omega$, there exists (at least) a basis consisting of its orthonormal ...
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1answer
56 views

Hermitian operator in an eigenbasis

Does "Hermitian operator in an orthonormal eigenbasis" mathematically translate to, $$\sum_{i} \omega_{i} |i\rangle \langle i|$$ Where $\omega_{i}$ is an eigenvalue and $|i\rangle$ is a normalized ...
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2answers
94 views

The Eigenvalue problem

In page 32 of R. Shankar's Principles of Quantum Mechanics is given the eigenvalue problem: We begin by rewriting Eq. (1.8.2)as $$(\Omega - \omega I)|V\rangle =|0\rangle \tag{1.8.3}$$ Opening ...
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1answer
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Why does the $s$ (and $m$) from the eigenstate $\big |s,m \big >$ come outside of the state and into the eigenvalue along with $\hbar$?

In my quantum mechanics class I've learned a notation for basis states which is $\big|s,m\big>$. From what I understand s is the spin (so if a particle was a spin $3\over 2$ it would always be in ...
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42 views

Eigenvalues of operators on a hyperbolic spacetime

In general, one can find the eigenvalues of the Laplace-Beltrami operator $\nabla^2$ on a hyperbolic space by taking the Selberg trace. Is there some analogous construction for calculating the ...
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1answer
56 views

Measurement of position in quantum mechanics

I know that when you perform a measurement of position in quantum mechanics, the wave function collapses to something proportional to it, but in a small range of values of positions, depending on the ...
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0answers
30 views

Eigenvectors and Eigenvalues of $a^\dagger a$ [duplicate]

A common question I have seen on problems sets and on exam papers is as follows (own words): Show that $a^\dagger$ and $a$ satisfy the canonical commutation relations. Hence show write down the ...
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1answer
94 views

Why can't the momentum eigenvalue of the momentum operator be imaginary or a function of $x$?

Say I want to test if the following function is an eigenfunction of the momentum operator: $$\psi(x,t) = A\exp{(-\alpha x^2})$$ $$\hat p [\psi(x,t)]= -i \hbar \frac{d}{dx}[A\exp{(-\alpha x^2})]$$ $$...
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Eigenvalue equation for interacting Green's functions

Studying the articles "Topological Hamiltonian as an exact tool for topological invariants" (https://arxiv.org/abs/1207.7341) and "Simplified Topological Invariants for Interacting Insulators" (https:/...
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59 views

Wave function collapse, when spectrum is continuous

Is known that when measuring position the possible measurement values are eigenvalues of the position operator. Before measurement the particle is supposed to be in a physical state (wave function) ...
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1answer
32 views

is it possible to have different states other than just one in Dirac Delta Potential?

Let's say, initially the state is in first excited state of finite well potential and then I change the width & depth of the well, eventually to Dirac delta potential, then what happens to the ...
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2answers
86 views

Spectral theorem and infinite square well

I was reading about the solution to the infinite square well and it imposes boundary conditions that make the wavefunction 0 outside the well. So, that means that physically any initial state is a ...
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95 views

Completeness relation for continuous spectrum

I was reading my lecture notes and I found this: I wanted to know if equation 4.2 always hold true for continuous spectrum. Meaning when you have continuous spectrum of self adjoint operators, in ...
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0answers
107 views

Dirac-delta orthonormality

Is my understanding that eigenfunctions of position operator and momentum operator exhibit, Dirac-delta orthonormality. I wanted to ask if all self-adjoint operators in quantum mechanics exhibit this ...
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0answers
57 views

Quantum Well Eigenvalues - Newton-Raphson

The eigenvalues of a quantum well must obey: $$\tan \left( \sqrt{\frac{a^2 m_e^* E_n}{2 \hbar^2}} \right) = \sqrt{\frac{(V_b-E_n)}{E_n}}$$ and $$\cot \left( \sqrt{\frac{a^2 m_e^* E_n}{2 \hbar^2}} \...