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expected value
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Definition/Summary
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| A short introduction to expectation value is given, both for discrete and continuous cases. |
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Equations
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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] |
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Scientists
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| Some mathematician from 19th century. |
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Recent forum threads on expected value
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Breakdown
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Physics
> Quantum
>> Fundamental Concepts
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Extended explanation
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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] |
Commentary
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