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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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How do I get exact energy eigenvalues for a quartic oscillator with $H=\frac{1}{2}p^2+g \, x^4$? [duplicate]

Can someone help me with solving the non-linear oscillator case with hamiltonian as follows: $$H=\frac{1}{2}p^2+g \, x^4$$
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98 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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101 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
71 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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How are energy eigenvalues of a wave-function related to the eigenvalue of matrices?

Schrodinger equation is a second order DE. Given a potential, I wish to perform standard -find the ground state energy, plot the wavefunction, find the reduced mass exercises using finite difference ...
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115 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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65 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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55 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 ...
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17 views

Normalizable vs non-normalizable startionary states and continuous vs discrete index [duplicate]

The non-normalizable stationary states (e.g., energy eigenstates of free particle) always labelled by continuous indices ($k$, in my example) while normalizable stationary states (such as the energy ...
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1answer
62 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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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
74 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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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
37 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
61 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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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
25 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
60 views

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
159 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
50 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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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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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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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
50 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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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
65 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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51 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
31 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
78 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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54 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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70 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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47 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}} \...
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65 views

1D delta potential hamiltonian

I have two concerns regarding the delta potential hamiltonian: Is my understanding that in quantum mechanics we use self-adjoint operators. However I cannot figure it out if the hamiltonian that ...
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Schrödinger equation on a lattice

Consider the following problem: The given Hamiltonian is: $$ H=-J\sum_{m,n}\left( e^{-i\phi n}a^{\dagger}_{m+1,n}a_{m,n}+a^{\dagger}_{m,n+1}a_{m,n} +h.c.\right)$$ Inserting this Hamiltonian in the ...
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1answer
68 views

Real eigenvalues of continuum spectrum of a self-adjoint operator

Is my understanding that if you assume eigenvectors of a self-adjoint operator are in Hilbert space, then is easy to prove that the eigenvalues must be real. However, it could happen that the "...
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27 views

Is it possible to obtain quasiparticle spectrum from standard Hamiltonian spectrum?

In many papers one can finds quasiparticle energy $\mathcal E_n$ spectrum for BCS like Hamiltonian $\hat H$ (with eigenvalues $E_n$), which is ussualy obtained from diagonalisation of transformed ...
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1answer
65 views

What can be said about the spectrum of a Hamiltonian of a single particle confined in a box with periodic boundary conditions?

I am coming up with that question as I simply cannot satisfy myself with the frustrating fear that it might not be possible to show that a Hamiltonian corresponding to a particle in a box with ...
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1answer
53 views

Type-I seesaw: order of magnitude of the eigenvalues of effective $M_\nu$

Consider a square matrix $C$ constructed out of two other square matrices $A$ and $B$ as $$C=-A^TB^{-1}A.$$ Suppose all the elements of $B$ are very large compared to those of $A$. In such a case, is ...
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Bloch's Theorem without periodic boundary condition - mathematically rigorous way

I am looking for a proof of Bloch's Theorem which does not use periodic boundary conditions. Sometimes one happens to see non-rigorous demonstrations of Bloch's Theorem without the use of periodic ...
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25 views

Periodic potential in 3 dimensions - Dimensionality of generalized eigenspaces of Hamiltonian?

So I was having a look at a single-particle Hamiltonian with a periodic potential in 3 dimensions. It is generally known that there are no bound states and the spectrum of H is purely continuous. As ...
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What is the eigenstate of the scalar product of two momenta applied on momentum eigenstate

Let's suppose a collision between some particle and a nucleus with $A$ nucleons. This particle has momentum $\vec{k}$ then $\vec{k}'$, with $\vec{k}'-\vec{k}=\vec{q}$ and let's say that the nucleon ...
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2answers
128 views

Eigenvectors and eigendecomposition of Pauli matrices, why isn't there many?

Say we are finding eigenvectors of $\sigma _z$, the eigenvalues are $1,-1$ so filling into the eigenvalue equation $\sigma _z (a,b)=(a,-b)=1(a,b)$ and we find that $b=0$. I am confused about why we ...
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245 views

Eigenvectors and eigenvalues of a metric

Suppose we have an $n$-dim Riemannian space $V_n$ endowed with a metric. More precisely, it is defined a $(0,2)$-type tensor field in $V_n$: $$g_{\mu\nu} = g_{\mu\nu}(x),\quad x\in V_n$$ Questions. ...
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1answer
225 views

Quantum pure quartic oscillator

It was recently brought to my attention, that there exist analytic solutions for the quantum pure-quartic oscillator with the hamiltonian $$ \hat{H} = \frac{1}{2m} \hat{p}^2 + \frac{\lambda}{24} \hat{...
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1answer
46 views

Uniqueness of simultaneous eigen states of two linear operators

I was solving a homework problem where the question gives the representation of two operators in matrix form, in some arbitrary set of basis vectors. It then asks to find the simultaneous eigen states ...
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2answers
50 views

Probability of a specific energy state

We consider the normalized wave function: $$\psi(x,t) = \sqrt{\frac{2}{3}}\psi_0(x)\exp\left(\frac{-iE_0t}{\hbar}\right) + \sqrt{\frac{1}{3}}\psi_1(x)\exp\left(\frac{-iE_1t}{\hbar}\right) $$ To ...