- #1
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Definition/Summary
A short introduction to expectation value is given, both for discrete and continuous cases.
Equations
For discrete probability distributions,
<Q> \ = \ \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
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:
<Q> = \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 - <A>)^2 p_n ,
\sum _n p_n = 1 ,
<A> = \sum _n A_nP_n ,
<<A>> = <A>
<A> is just a number, we can thus show that:
(\Delta A ) ^2 = <A^2> + <A>^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 <D>, of the outcome of rolling dice:
<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}
since each value has the equal probability of 1/6 .
CONTINUOUS DISTRIBUTIONS:
Now this was for the discrete case, in the continuous 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 - <Q>)^2> = <Q^2> - <Q>^2
* 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!
A short introduction to expectation value is given, both for discrete and continuous cases.
Equations
For discrete probability distributions,
<Q> \ = \ \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
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:
<Q> = \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 - <A>)^2 p_n ,
\sum _n p_n = 1 ,
<A> = \sum _n A_nP_n ,
<<A>> = <A>
<A> is just a number, we can thus show that:
(\Delta A ) ^2 = <A^2> + <A>^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 <D>, of the outcome of rolling dice:
<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}
since each value has the equal probability of 1/6 .
CONTINUOUS DISTRIBUTIONS:
Now this was for the discrete case, in the continuous 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 - <Q>)^2> = <Q^2> - <Q>^2
* 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!
