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What is expected value

  1. Jul 24, 2014 #1
    Definition/Summary

    A short introduction to expectation value is given, both for discrete and continuous cases.

    Equations

    For discrete probability distributions,

    [tex] <Q> \ = \ \sum _n Q_n p_n[/tex]

    For continuous distributions specified by a normalized, real space wave-function [itex]\psi(x)[/itex],

    [tex]< Q > = \int _{\text{All space}} \psi^*(x)Q(x)\psi(x) dx [/tex]

    Extended explanation

    NOTATION:

    The notation < > comes from statistics, so it is a general notation which QM scientists borrowed.


    DEFINITIONS:

    The expectation value of an observable associated with an operator [itex]Q[/itex] is defined as:

    [tex] <Q> = \sum _n Q_n p_n [/tex]

    in the case of a discrete spectrum, where [itex]Q_n[/itex] is the eigenvalue of Q for a state labeled by the index n, and [itex]p_n[/itex] is the probability of measuring the system in this state.


    DISCRETE DISTRIBUTIONS:

    Variance in statistics, discrete case:
    [tex](\Delta A ) ^2 = \sum _n (A_n - <A>)^2 p_n ,[/tex]
    [tex] \sum _n p_n = 1 ,[/tex]
    [tex] <A> = \sum _n A_nP_n ,[/tex]
    [tex] <<A>> = <A> [/tex]
    [itex] <A> [/itex] is just a number, we can thus show that:
    [tex] (\Delta A ) ^2 = <A^2> + <A>^2 [/tex]
    and
    [tex]<(\Delta A ) ^2> = (\Delta A ) ^2. [/tex] as an exercise, show this.

    where [itex] \sum _n p_n = 1 [/itex] and [itex] A_n [/itex] is the outcome of the n'th value.


    EXAMPLE:

    As an exercise, let's find the expectation value <D>, of the outcome of rolling dice:

    [tex] <D> = 1 \cdot \dfrac{1}{6} + 2 \cdot \dfrac{1}{6} + 3 \cdot \dfrac{1}{6} + 4 \cdot \dfrac{1}{6} + 5 \cdot \dfrac{1}{6} + 5 \cdot \dfrac{1}{6} = \dfrac{7}{2}[/tex]
    since each value has the equal probability of [itex] 1/6 .[/itex]


    CONTINUOUS DISTRIBUTIONS:

    Now this was for the discrete case, in the continous case:
    [tex]< Q > = \int _{\text{All space}} f(x)Q(x)f(x) dx [/tex]
    where [itex] f^2(x) [/itex] is the probability density distribution : [itex] \int f^2(x) dx = 1 [/itex].

    That was if the probability density distribution is real, for complex valued (such as quantum mechanical wave functions):
    [tex]< Q > = \int _{\text{All space}} \psi^*(x)Q(x)\psi(x) dx [/tex]
    [itex] \int |\psi (x)|^2 dx = 1 [/itex].

    EXAMPLES:

    Position:
    [tex]< x > = \int _{\text{All space}} \psi^*(x)x\psi(x) dx = \int x|\psi (x)|^2 dx [/tex]

    Momentum:
    [tex]< p > = \int _{\text{All space}} \psi^*(x)(-i\hbar\dfrac{d}{dx})\psi(x) dx [/tex]

    Now the variance is:
    [tex] \Delta Q ^2 = <(Q - <Q>)^2> = <Q^2> - <Q>^2 [/tex]

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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