1
vote
2answers
65 views

Eigenvalues being physical observables

I think I'm comfortable with the PDE solutions to the Schrodinger equation. But as soon as we start putting these values in a matrix (in dirac notation), I lose my understanding and everything ...
1
vote
0answers
71 views

An Operator Identity relating to Trace [duplicate]

Suppose that $\hat H$ is an operator (typically a Hamiltonian) and $\beta$ is a positive parameter (typically $\beta=1/k_BT$). Show that $$ \mathbf{Tr}\Big(e^{-\beta\hat H}\Big) \geq ...
1
vote
2answers
47 views

How does Dirac conclude that $X_r(c_r)$ cannot vanish?

On page 32 of Dirac's book Principles of Quantum Mechanics, he considers the case when the linear, Hermitian$^1$ operator $\xi$ satisfies an algebraic equation $$\phi(\xi)\equiv(\xi - c_1)(\xi - ...
2
votes
1answer
64 views

Approach to expressing $|n\rangle\langle n| $ as a polynomial when eigenvalues are degenerate?

If ${|n\rangle}$ are eigenvectors of an operator $A$ then $|n\rangle\langle n| $ can be expressed in terms of a finite order polynomial $$|n\rangle\langle n| =\prod_{m\ne n} \frac{A-a_m}{a_n-a_m}$$ ...
2
votes
1answer
47 views

Eigenfunctions corresponding to degenerate spectra

It is well knwon that an eigenstate can be obtained by superposing wavepackets. In other words, if $\Psi({\bf x},t)$ is the solution of the time dependent Schroedinger equation for an initial ...
0
votes
1answer
25 views

How to write “postselection” operator?

Suppose, I wish to know an operator, which eigenvalue is 1 if state is exactly F and 0 ...
1
vote
1answer
59 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 ...
1
vote
1answer
44 views

Symmetric eigenfunctions?

So a symmetric eigenfunction / wavefunction is defined as: $$P_{ij} ψ_a (r_1,r_2,…,r_i,…,r_j,…,r_N )=ψ_a(r_1,r_2,…,r_i,…,r_j,…,r_N )$$ But for it to be symmetric does this have to be true for all $ij$ ...
7
votes
0answers
118 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 ...
4
votes
1answer
88 views

Is there some quantum potential producing exponential eigenvalues?

Usual central potentials produce quantum spectra with energy levels going as $n$, $n^2$, $n^3$ and so on, being $n$ the quantum number of the orbit. In the other extreme we have "dirac-delta" ...
1
vote
1answer
63 views

Expectation value and Dispersion of an Operator

Suppose we have an operator $Q$ with eigenvalue $q$. Expectation value is $\langle Q \rangle$ and dispersion $D(Q) = \sqrt{\langle \left( Q - \langle Q \rangle \right)^2 \rangle} $. I want to find ...
1
vote
0answers
57 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 ...
0
votes
0answers
66 views

What is the eigenvalue of $J_z$?

In the calculation of the Zeeman Effect, the most important calculation is $$\langle J_z + S_z\rangle.$$ Suppose we want to find the Zeeman Effect for $(2p)^2$, meaning $l = 1$. In Sakurai's book, ...
1
vote
1answer
88 views

Normalizing continuous eigenstates

As far as I understand, to normalize the eigenfunctions, corresponding to the continuous spectrum, we use Dirac delta function: $\langle \psi_\lambda \mid \psi_{\lambda'} \rangle = \delta(\lambda - ...
0
votes
0answers
42 views

Showing that the maximum possible uncertainty for any observable is half the difference between its maximum and minimum eigenvalues

Show that the maximum possible uncertainty for any observable is $\frac{1}{2}|x_2 - x_1|$ where $x_1$ and $x_2$ are the extreme eigenvalues of X (Maximize $\Sigma_i p_ix_i^2 - (\Sigma_i p_ix_i)^2$) ...
1
vote
3answers
66 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 ...
0
votes
2answers
85 views

Momentum Operator in Quantum Mechanics

1) What is the difference between these two momentum operators: $\frac{\hbar}{i}\frac{\partial}{\partial x}$ and $-i\hbar\frac{\partial}{\partial x}$? How are these two operators the same? My ...
3
votes
2answers
186 views

Eigenenergies and eigenkets given the Hamiltonian

For a two level system the Hamiltonian is: $$ H=a(|1\rangle \langle1|-|2\rangle\langle2|+|1\rangle\langle2|+|2\rangle\langle1|) $$ where $a$ is a number with the dimension of an energy. I need to ...
6
votes
2answers
185 views

Eigenvalue problem for differential equations in QM

I have a very simple question with regard to numerical methods in physics. I want to solve the eigenvalue problem for a particle moving in an arbitrary potential. Let's take 1D to be concrete. I.e. I ...
3
votes
1answer
67 views

Eigenvalues of Infinite Dimensional Matrix [duplicate]

If I take a infinite-dimensional square matrix, what can I say about its eigenvalue spectrum? Will they have a discrete infinity of eigenvalues or continuous infinity of them?
4
votes
1answer
288 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 ...
0
votes
1answer
214 views

Proof of the time-independent Schrödinger equation

I have a question regarding the proof of the time-independent Schrödinger equation. So if we have a time-Independent Hamiltonian, we can solve the Schrödinger equation by adopting separation of ...
1
vote
1answer
158 views

Spin eigenvalues and eigenvectors problem. Is this the correct way to solve it?

An electron is described by the Hamiltonian $ H=\frac{e}{mc}\bar{S}\cdot\bar{B} $ where $\bar{S} =(S_x,S_y,S_z)$ is the spin operator and $\bar{B}$ the magnetic field. For $t>0$ ...
1
vote
1answer
96 views

Clarify formula in quantum perturbation theory

I'm studying perturbation theory in the context of quantum mechanics. My lecture notes say that in order to calculate the first-order correction of eigenfunction $\psi_n$, that is $\psi_n^{(1)}$, I ...
12
votes
2answers
341 views

Eigenfunctions of the Runge-Lenz vector

The hamiltonian for the hydrogen atom, $$ H = \frac{\mathbf{p}^2}{2m} - \frac{k}{r} $$ is spherically symmetric and it therefore commutes with the angular momentum $\mathbf{L}$; this causes all its ...
1
vote
2answers
204 views

The eigenspinors for the spin operator in the $x$-direction?

$$S_x= \frac{\hbar}{2}\quad\begin{pmatrix}0&1\\1&0\end{pmatrix}\quad$$ $$S_x{X_+}^{(x)}=\frac{\hbar}{2}{X_+}^{(x)}$$ How can I find the eigenvalue of $S_x$? My book says $$ \left| ...
0
votes
1answer
313 views

Eigenvectors of the angular momentum operator $S_x$ [closed]

For a spin of $\frac{1}{2}$ the angular momentum operator can be written as $\vec{S} = \frac{\hbar}{2} \vec{\sigma}$ in matrix form. Find the eigenvalues and eigenvectors of $S_x$ where $\sigma_x = ...
2
votes
1answer
107 views

Product of position eigenvectors at different times

I've been thinking about this, and it might sound like a stupid question, but I can't seem to find an answer anywhere, here goes: Whenever we calculate expecation-values between two position ...
0
votes
1answer
342 views

Harmonic Oscillator Energy to Momentum Expectation Value

If we are given a wave function written in terms of harmonic oscillator energy eigenfunctions how can we determine the maximum possible momentum expectation value? It's a combination of the first two ...
2
votes
1answer
494 views

Why are orthogonal functions and eigenvalues/functions so important in quantum mechanics?

The mathematics and physics we have studied so far at university are heavily focused around the idea of orthogonal functions, orthogonality, sets of solutions, eigenvalues and eigenfunctions. Why ...
1
vote
1answer
160 views

Matrix operations on Quantum States in a composite quantum system

Intro (you may skip this if you're an expert, I'm including this for completeness): Say I have two bases for two systems, The first is a spin-1/2 system $|+\rangle = \left(\begin{array}{c} 1\\0 ...
14
votes
6answers
1k 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 ...
1
vote
1answer
101 views

Weird Behaviour of the act of measurement to a quantum system

I and my friend were disputing about some weird behaviour of the act of measuring some observables quantities e.g. Energy, position. But I still don't think what he said is strictly true. He said" ...
2
votes
1answer
193 views

Relationship between two eigenfunctions of the time-independent Schrödinger Equation in one dimension?

What is the relationship between two eigenfunctions of the time-independent Schrödinger Equation (in one spatial dimension) if they both have the same eigenvalue?
2
votes
2answers
242 views

Angular momentum of quantum system

Problem: A physical system is in the common eigenstate of $\hat{L^2}$ and $\hat{L_z}$. Calculate the following quantities: $\langle L_x\rangle,\langle L_y\rangle,\langle L_z\rangle,\langle L_x L_y + ...
0
votes
2answers
2k views

Expectation Values in Quantum Mechanics

Why is the expectation value what it is? Why don't you apply the operator, then multiply that by it's conjugate?
1
vote
2answers
177 views

Complex Versus Real Wave Velocities in Quantum Mechanics

There's a fantastic quote in Schrodinger's second 1926 paper1 that apparently provides some motivation for the discrete energy levels (I think) that I'm having trouble interpreting: I would not ...
2
votes
1answer
95 views

Is continuous evolution from one eigenstate of operator $O$ to another $O$-eigenstate possible?

Eigenvectors associated with distinct values of an observable are orthogonal, according to quantum mechanics. Does this entail that a quantum system cannot continuously evolve from one eigenstate ...
1
vote
1answer
434 views

Total angular momentum - single electron

I have been dealing with total angular momentum of the single electron which is outside the closed shells in which sum of the angular momentums is zero. My book says that total atomic angular ...
0
votes
0answers
233 views

Dirac Delta Potential and bound/scattered states

Why does the attractive Dirac Delta distribution (function) potential $V = \alpha\delta$(x) (for negative $\alpha$) yield both bound AND scattered states? Is this due to the definition of the Dirac ...
7
votes
4answers
1k views

Bounded and Unbounded (Scattering) States in Quantum Mechanics

I understand that bounded states in quantum mechanics imply that the total energy of the state, $E$, is less than the potential $V_0$ at + or - spatial infinity. Similarly, the scattering state ...
1
vote
3answers
2k 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 ...
1
vote
2answers
2k views

The Energy Eigenvalue of a Wavefunction

I've been reading an introduction to quantum mechanics online, and while constructing the Schrodinger equation for a free particle, the equation $i\hbar \frac{d \Psi}{dt}=\hbar\omega\Psi$ is obtained. ...
2
votes
0answers
147 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 ...
-1
votes
1answer
163 views

Wavefunction operators and the observable [closed]

So I got this from the exam I had yesterday. I couldn't really answer it other and it played on my mind through the night Show that if a wave function $\psi$ , is an eigenfunction of an operator [Q], ...
-2
votes
2answers
236 views

Determine whether the ground state is an eigenfunction of [p] and of [p^2] [closed]

Consider a particle confined in an infinite square well potential of width L, $$V(x)=\left\{ \begin{array}{ll}\infty, &{\rm for}\ (x \le 0)\vee (x \ge L) \\0, &{\rm for} \ 0 < x < L ...
3
votes
1answer
489 views

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 ...
5
votes
3answers
985 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, ...
0
votes
1answer
80 views

Eigenvalue $a_n$

Q1: In Zetilli's book page 166 (ch. "Postulates of QM", eq. 3.1) i encountered an expression $\hat{A}|\psi\rangle = a_n|\psi_n\rangle$. I know this is an eigenvalue equation, but i have seen another ...
2
votes
2answers
382 views

How do we know that $\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue $W$?

I am kind of new to this eigenvalue, eigenfunction and operator things, but I have come across this quote many times: $\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue $W$. ...