If theta is a constant 1/pi from 0 to pi then:

1. The expectation value of theta <theta> = pi/2
2. <cos(theta)> = 0 which makes sense
3. <sin(theta)> = 2/pi
4. The standard deviation = pi/(2*sqrt(3))

I don't understand 3 and 4. Why isn't the expectation value of the sin of theta equal to 1? And what does a standard deviation of a constant even mean?

I know how to calculate them, I just don't understand them.

If theta is a constant 1/pi from 0 to pi then:

1. The expectation value of theta < theta > = pi/2
2. < cos(theta) > = 0 which makes sense
3. < sin(theta) > = 2/pi
4. the standard deviation = pi/(2*sqrt(3))

I don't understand 3 and 4. Why isn't the expectation value of the sin of theta equal to 1? And what does a standard deviation of constant even mean?

Originally posted by Sacroiliac
If theta is a constant 1/pi from 0 to pi then:

1. The expectation value of theta < theta > = pi/2
2. < cos(theta) > = 0 which makes sense
3. < sin(theta) > = 2/pi
4. the standard deviation = pi/(2*sqrt(3))

I don't understand 3 and 4. Why isn't the expectation value of the sin of theta equal to 1? And what does a standard deviation of constant even mean?

well, i don t know what #4 means, are you sure you are reading the problem correctly? standard deviation of what function? what variable?

but here is the answer for #3

&lang;sin &theta;&rang;=1/&pi;&int;sin &theta; d&theta; = 1/&pi;(1--1)=2/&pi;

Perhaps you mean:

f(t) = 1/pi

In which case:

&lt;f(t)&gt; = integral (0->pi) t f(t) dt = pi/2

&lt;sin(f(t))&gt; = integral (0->pi) t sin(1/pi) dt = pi²/(2 sin(1/pi))

Originally posted by lethe
well, i don t know what #4 means, are you sure you are reading the problem correctly? standard deviation of what function? what variable?

but here is the answer for #3

&lang;sin &theta;&rang;=1/&pi;&int;sin &theta; d&theta; = 1/&pi;(1-1)=2/&pi;

Thanks lethe but I knew how to calculate it I just don't understand what it means.

The problem as stated:

The needle on a broken speedometer is free to swing, and bounces perfectly off the pins at either end, so that if you give it a flick it is equally likely to come to rest at any angle between 0 and pi.

Then it asks you calculate the above. This is from Griffiths "Inroduction to Quantum Mechanics" page 10. I just finished Morrison's book and was looking through this one and when I worked this problem I said "Say What!"

BTW your integral signs don't show up in either netscape or Explorer. Do I need a newer release?

Originally posted by Sacroiliac
Thanks lethe but I knew how to calculate it I just don't understand what it means.

The problem as stated:

The needle on a broken speedometer is free to swing, and bounces perfectly off the pins at either end, so that if you give it a flick it is equally likely to come to rest at any angle between 0 and pi.

Then it asks you calculate the above. This is from Griffiths "Inroduction to Quantum Mechanics" page 10. I just finished Morrison's book and was looking through this one and when I worked this problem I said "Say What!"

BTW your integral signs don't show up in either netscape or Explorer. Do I need a newer release?

i changed the font. did that help? the browser i am on is MozillaFirebird, one of the descendants of the netscape legacy. if you are using a newish version of netscape (7.0 or so), then you should be seeing the exact same thing that i see. maybe your system just lacks a lot of fonts.

anyway, i have a copy of griffiths. i am looking at it now

here are the integrations you should do for part b:

&lang;&theta;&rang;=&int;&theta;&rho;(&theta;)d&theta;
&lang;&theta;²&rang;=&int;&theta;²&rho;(&theta;)d&theta;
&sigma;²=&lang;&theta;²&rang;-&lang;&theta;&rang;²

James R and lethe, I know how to calculate the answers, I just don't know how to interpret them.

If the expectation value of theta is equal to pi/2 then why isn't the expectation value of the sign equal to 1. The expectation value of the cos is equal to 0 just as you'd expect. And what does the standard deviation of a function that's equal to a constant even mean?

lethe:

You're using HTML entities which are not supported by many browsers (IE, Netscape). They simply appear as squares on my screen. It's not a font issue.


Sacroiliac:

<i>If the expectation value of theta is equal to pi/2 then why isn't the expectation value of the sign equal to 1.</i>

The functions are defined from zero to Pi. Over that range, the cosine function is equally as often positive as negative, whereas sine is always positive. Hence, when you combine these functions with a constant probabilty distribution, you get the results you get.

<i>And what does the standard deviation of a function that's equal to a constant even mean?</i>

It's the standard deviation of the possible values of theta which the problem is talking about. The standard deviation gives a measure of the spread of values expected from theta.

Thanks J R I'll think about that for awhile and see if I can absorb it. But right now I'm going to bed as I should have about an hour ago.

Originally posted by James R

Sacroiliac:


The functions are defined from zero to Pi. Over that range, the cosine function is equally as often positive as negative, whereas sine is always positive. Hence, when you combine these functions with a constant probabilty distribution, you get the results you get.



Hmmm, good answer. I was stuck in the thought process that the expectation value of the sin of theta should be equal to the sin of the expectation value of theta.

< sin(theta) > = sin(< theta >)

Ah well, maybe I should study art appreciation. Or better yet -- beer appreciation.

Originally posted by Sacroiliac
function that's equal to a constant even mean?

theta is not a constant. it ranges from 0 to pi.

Originally posted by James R
lethe:

You're using HTML entities which are not supported by many browsers (IE, Netscape). They simply appear as squares on my screen. It's not a font issue.

hmm...

well, i really like using HTML mathematical symbols in my posts. these HTML codes are as old as HTML itself, and there is no excuse for microsoft not to include them in IE.

as for netscape, well, if you are using netscape 4, i have no pity for you, and i don t believe you if you say that netscape 7 doesn t support these symbols.

call me a zealot, but i think if the tool doesn t work, you shouldn t use it. and there are alternatives out there.

Originally posted by lethe
hmm...


as for netscape, well, if you are using netscape 4, i have no pity for you, and i don t believe you if you say that netscape 7 doesn t support these symbols.



lethe I just fired up netscape 7 and it shows your integral signs ok, but your expectation symbols (or whatever you call these things < > ) show up as question marks.