Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: how to find the expectation value of cos x

  1. Dec 7, 2018 #1
    1. The problem statement, all variables and given/known data
    If x is a continuous variable which is uniformly distributed over the real line from x=0 to x -> infinity according to the distribution f (x) =exp(-4x) then the expectation value of cos 4x is?

    Answer is 1/2
    Follow· 01
    Request




    2. Relevant equations
    the expectation value of any function g is, <g>=∫Ψ*gΨdx

    3. The attempt at a solution

    Here,
    Ψ=f(x)
    since f is independent of time
    ∴Ψ*=f(x)
    now, for <cos 4x>=∫exp(-4x)cos(4x)exp(-4x)dx
    let <cos4x>=I=∫cos(4x)exp(-8x)dx
    rest of the solution if in the picture attached
    my main doubt is how to put the limit infinity in sin and cos
     

    Attached Files:

    Last edited by a moderator: Dec 7, 2018
  2. jcsd
  3. Dec 7, 2018 #2

    Delta2

    User Avatar
    Homework Helper
    Gold Member

    The limits ##\lim_{x\rightarrow \infty}\cos{4x}##, ##\lim_{x\rightarrow \infty}\sin{4x}## do not exist, however here you have the limits

    ##\lim_{x\rightarrow \infty}e^{-8x}\cos{4x}## , ##\lim_{x\rightarrow \infty}e^{-8x}\sin{4x}## which are zero, because ##lim_{x\rightarrow \infty}e^{-8x}=0## and the functions ##\sin{4x}, \cos{4x}## are bounded. (It is a well known lemma that the limit of the product of a bounded function with another function that has limit zero, is also zero).
     
  4. Dec 7, 2018 #3

    ehild

    User Avatar
    Homework Helper

    The probability distribution function is not the same as the wavefunction in QM.
     
  5. Dec 8, 2018 #4

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    This does not read like a quantum theory question to me, just straight probability. But the given f(x) is not a valid pdf; its integral over the full range is not 1. Seems like it should be 4e-4x.
    Anyway, to answer your question, all you care about at infinity is that the trig functions are bounded. Look at the other factor.

    Beaten by Δ2 and ehild because I spent time figuring out the answer, and I do not believe it is 1/2. Also, there is still the problem that the given pdf is not valid.

    Edit: the reason I got a different answer is that I used cos(x), as stated in the title, instead of cos(4x), as stated in post #1.
     
    Last edited: Dec 8, 2018
  6. Dec 8, 2018 #5

    Delta2

    User Avatar
    Homework Helper
    Gold Member

    That was my concern also, but my subconscious thought was that he meant wave function, though the problem statement clearly states probability density function.

    So the initial integral to start with would be ##I=\int_0^{\infty}4e^{-4x}\cos{4x}dx##.
     
  7. Dec 8, 2018 #6

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Except, as I posted, it fails the totality of 1 criterion.
     
  8. Dec 8, 2018 #7

    Delta2

    User Avatar
    Homework Helper
    Gold Member

    Yes it should have been with a factor 4 in front (as you already said), probably a typo by the OP.
     
  9. Dec 8, 2018 #8

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You may find it a bit simpler to spot that a likely integral of ##e^{-\lambda x}\cos(\mu x)## is the form ##e^{-\lambda x}(A\cos(\mu x)+B\sin(\mu x))## then differentiate that to find A and B.
     
  10. Dec 8, 2018 #9
    Could you please explain or suggest a source from which I can understand the difference.
     
  11. Dec 8, 2018 #10
    It is possible.
    The above question is from a previous year entrance test.
     
  12. Dec 8, 2018 #11
    Isn't the expectation value supposed to be ∫f* cos4x f dx
     
  13. Dec 8, 2018 #12
    I have posted the question just as it was given.
     
  14. Dec 8, 2018 #13

    Delta2

    User Avatar
    Homework Helper
    Gold Member

    That is correct if ##f## is given as a wave function. However the problem states that ##f## is a probability density function hence the expectation value of the variable ##g(x)## is simply ##\int g(x)f(x)dx##.

    To unify the two cases, when ##f## is the wave function then the probability density function is ##|f|^2=f^{*}f##
     
    Last edited: Dec 8, 2018
  15. Dec 8, 2018 #14
    this gives the correct answer
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted