How to show that expectation value is always positive?

ngkamsengpeter
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Homework Statement


In quantum mechanics, how to show that the expectation value is always positive?



Homework Equations





The Attempt at a Solution

 
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Can you be more specific? Because as you stated it, one can easily find a counter-example (e.g. <x> for a Gaussian wavepacket centered at some x0 < 0).
 
CompuChip said:
Can you be more specific? Because as you stated it, one can easily find a counter-example (e.g. <x> for a Gaussian wavepacket centered at some x0 < 0).
Questions just tell us to show that A+A is always self-adjoin and then show that expectation is non-negative.
 
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Thread 'Need help understanding this figure on energy levels'

This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary to introduce the principal quantum number, ##n##, which simply orders the allowed energies, starting with 1 for the ground state. (see the figure)" I still don't understand the figure :( Here is...
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Thread 'Understanding how to "tack on" the time wiggle factor'

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The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 = \lambda, \mathbf{v}_2 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix} \sigma_3 = -\lambda, \mathbf{v}_3 = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \\ 0 \end{pmatrix} $$ There are two ways...
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