GREPhysics.NET: GR0177 #27

GR0177 #27

Problem

GREPhysics.NET Official Solution

This problem is still being typed.

Quantum Mechanics $\Rightarrow$ }Hermitian Operator

The eigenvalues of a Hermitian operator are always real. This is one of those quantumisms one memorizes after even a semi-decent course in QM.

One can quickly prove it by the other quantumism-proof one memorized for that same course: Suppose one has $A|\psi\rangle = a |\psi\rangle$ . Then, $A^\dag |\psi\rangle = a^* |\psi\rangle$ .

$A | \psi \rangle - A^\dag | \psi \rangle = (a-a^*)|\psi\rangle$ . But, the left side is just 0 from, as $A^\dag = A$ from the definition of Hermitian operator. Then, the right side must be 0, too, since $\langle \psi |\psi \rangle \neq 0$ in general.

Alternate Solutions

ramparts
2009-10-06 18:07:11

Hermitian operators represent observables. Their eigenvalues are things we measure. It is very hard to measure imaginary things :P

Reply to this comment

Comments

chemicalsoul
2009-10-28 10:02:53

The dot product of orthogonal vectors should be equal to zero which gives x =5. Tell me if i am wrong.

Reply to this comment

ramparts
2009-10-06 18:07:11

Hermitian operators represent observables. Their eigenvalues are things we measure. It is very hard to measure imaginary things :P

ETScustomer
2017-10-08 20:59:05

I cannot even imagine how hard it is!

poopterium
2017-10-26 02:05:27

booooooooooooooooooo

Reply to this comment

Post A Comment!

Bare Basic LaTeX Rosetta Stone
LaTeX syntax supported through dollar sign wrappers $, ex., $\alpha^2_0$ produces $\alpha^2_0$ .
type this...	to get...
$\int_0^\infty$	$\int_0^\infty$
$\partial$	$\partial$
$\Rightarrow$	$\Rightarrow$
$\ddot{x},\dot{x}$	$\ddot{x},\dot{x}$
$\sqrt{z}$	$\sqrt{z}$
$\langle my \rangle$	$\langle my \rangle$
$\left( abacadabra \right)_{me}$	$\left( abacadabra \right)_{me}$
$\vec{E}$	$\vec{E}$
$\frac{a}{b}$	$\frac{a}{b}$

The Sidebar Chatbox...
Scroll to see it, or resize your browser to ignore it...

All-Solutions List | Latest Comments | Unanswered Questions (Cries for Help!) |::| About

This site is not affiliated with ETS, nor is it officially endorsed (yet) by ETS. This site is the work of an individual named Yosun Chang. The solutions, sans user comments and ETS questions, are © 2005 by Yosun Chang except where noted. All rights reserved. / The comments appending particular solutions are due to the respective users. / Educational Testing Service, ETS, the ETS logo, Graduate Record Examinations, and GRE are registered trademarks of Educational Testing Service. The examination questions are © 1997 and © 2001 by ETS.

Bare Basic LaTeX Rosetta Stone