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expected value

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

 Equations For discrete probability distributions, $$\ = \ \sum _n Q_n p_n$$ For continuous distributions specified by a normalized, real space wave-function $\psi(x)$, $$< Q > = \int _{\text{All space}} \psi^*(x)Q(x)\psi(x) dx$$

 Scientists Some mathematician from 19th century.

 Recent forum threads on expected value

 Breakdown Physics > Quantum >> Fundamental Concepts

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 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 $Q$ is defined as: $$= \sum _n Q_n p_n$$ in the case of a discrete spectrum, where $Q_n$ is the eigenvalue of Q for a state labeled by the index n, and $p_n$ is the probability of measuring the system in this state. DISCRETE DISTRIBUTIONS: Variance in statistics, discrete case: $$(\Delta A ) ^2 = \sum _n (A_n - )^2 p_n ,$$ $$\sum _n p_n = 1 ,$$ $$= \sum _n A_nP_n ,$$ $$<> =$$  is just a number, we can thus show that: $$(\Delta A ) ^2 = + ^2$$ and $$<(\Delta A ) ^2> = (\Delta A ) ^2.$$ as an exercise, show this. where $\sum _n p_n = 1$ and $A_n$ is the outcome of the n'th value. EXAMPLE: As an exercise, let's find the expectation value , of the outcome of rolling dice: $$= 1 \cdot \dfrac{1}{6} + 2 \cdot \dfrac{1}{6} + 3 \cdot \dfrac{1}{6} + 4 \cdot \dfrac{1}{6} + 5 \cdot \dfrac{1}{6} + 5 \cdot \dfrac{1}{6} = \dfrac{7}{2}$$ since each value has the equal probability of $1/6 .$ CONTINUOUS DISTRIBUTIONS: Now this was for the discrete case, in the continous case: $$< Q > = \int _{\text{All space}} f(x)Q(x)f(x) dx$$ where $f^2(x)$ is the probability density distribution : $\int f^2(x) dx = 1$. That was if the probability density distribution is real, for complex valued (such as quantum mechanical wave functions): $$< Q > = \int _{\text{All space}} \psi^*(x)Q(x)\psi(x) dx$$ $\int |\psi (x)|^2 dx = 1$. EXAMPLES: Position: $$< x > = \int _{\text{All space}} \psi^*(x)x\psi(x) dx = \int x|\psi (x)|^2 dx$$ Momentum: $$< p > = \int _{\text{All space}} \psi^*(x)(-i\hbar\dfrac{d}{dx})\psi(x) dx$$ Now the variance is: $$\Delta Q ^2 = <(Q - )^2> = - ^2$$

Commentary

 tiny-tim @ 05:14 AM Dec10-10 Fixed missing LaTeX. No other changes.

 Gokul43201 @ 09:55 AM Feb10-09 Extended equation section to include continuous distributions, and amended the explanation a little, and made correction to variance (it was missing an outer pair of < >). More later.

 Redbelly98 @ 07:16 AM Feb9-09 Here's another issue: If f(x) is a probability density, then∫ f(x) dx = 1and it's not f2(x) in the integrand. Also, = ∫ Q(x) f(x) dx(not f(x) Q(x) f(x) in the integrand) I agree with tim, expectation value was always used in my quantum mechanics course. The first part of this entry would be a good item for the Mathematics, Probability/Stats section. Perhaps it should be moved there? Then create a separate "expectation value" entry with the quantum mechanics in it?

 tiny-tim @ 01:53 PM Feb8-09 title: wikipedia regards "expected value" as being from ordinary mathematical probability-theory, and "expectation value" as having a specifically physical quantum-theory meaning: this seems right to me, so perhaps we should have two entries, one for each concept? (and the dice example is definitely not physics ) (btw, on a forum search "expected value" gets 336 hits, and "expectation value" gets 500 hits back to November '05)

 malawi_glenn @ 01:50 PM Feb7-09 great! I forgot to decide what to put there, so I wrote "?" meanwhile ;-)

 Redbelly98 @ 07:45 AM Feb7-09 EDITS:Changed title from "The Expected Value." to "expected value". This will have more opportunities for autolinking in forum posts. (It would be nice if "expectation value" could also get autolinked to here somehow.) Included equation for discrete case in Equations section. This had been simply: = ? ----- Note: entry created Feb. 6, 2009 by IM4Strings

 malawi_glenn @ 02:10 AM Feb7-09 The formulation of this post was formulated as a question.. I changed that.