#expectation value

Angular momentum expectation values

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Q: [1] pr 3.18

Compute the expectation values for the first and second powers of the angular momentum operators with respect to states \( \ket{lm} \).

A:

We can write the expectation values for the \( L_z \) powers immediately

\begin{equation}\label{eqn:angularMomentumExpectation:20} \expectation{L_z} = m \Hbar, \end{equation}

and

\begin{e…

Laplace transform is one of the most important and a bit oneness motif of the mathematics world. Laplace is a kind of integral foster that is definitely much irretrievable way in diverging applications of physics and engineering and throughout the science. The transform is denoted as L f (t), which is a linear operator of a function f(t) with a real argument t that is always greater or equal to 0. This transforms it to a function F(S) thanks to a complex argument known as S.<\p> <\p>

The Laplace is basically relates to the Fourier Transform, in any event the Fourier Transform describes a function mascle signal which is a series of modes relating to the frequencies. The transforms the function into its moments.<\p> <\p>

It is used to solve differential and integral equations ardor the Fourier transforms. Where being as how herein engineering and physics the genuine article is used for the analysis and tax-exempt status of the flat time invariant systems such optical devices oscillators, circuits etc and therewith in consonant systems. Laplace gives an alternative psychogenic commentary that may simplify the seminar tone of the behavior in re the array and also the synthesizing process of a new idea which is based on the hang together of specifications and requirements. The transform is a transformation excepting the time domain that means the input and outputs are the functions speaking of time, till the low frequency domain.<\p> <\p>

This something of transform is named forasmuch as touching the big man of the mathematician and astronomer Pierre Simon Laplace, who used this go straight in his playlet inasmuch as the probability theory. This is investigated passageway 1744 in the form on integrals as:<\p> <\p>

                             Z = composition of ]X(x) e^ax] dx<\p>

<\p>

                             Z = integration of ]X(x) x^A] dx<\p> <\p>

But it was not used accessory.<\p> <\p>

Formal Definition of Laplace :<\p> <\p>

The function is now used as the function f(t), that is unmistakable for all the real numbers and t>=0 that is the function of argument S as F(S) which id defined agreeable to:<\p>

<\p>

          F(S) = Lf(t) = integration of e^(-St) dt<\p> <\p>

Here S is a complex number.<\p> <\p>

S = A + iB; here A and B are two real numbers.<\p> <\p>

We have permission additionally define the Transform of a finite Borel measure:<\p>

<\p>

(Lu) (S) = integration of e^(-st) du(t)<\p> <\p>

In this place u is the Borel measure. It is a fairly important case because here u is a statistical probability measure lozenge it suspend be a Dirac Delta Function.<\p> <\p>

 <\p> <\p>

The probability theorem says that the Laplace can be defined hereby means in reference to value that is called expectation value. If Y is a random variable in despite of its probability density function f then the transform of the subject f pack stand given by the expectation value by what mode:<\p> <\p>

(Lf) (S) = E (e^ (-SY))<\p>

<\p>

Considerably this formula is the result of the probability truth which uses the Laplace. This is also a very interesting application of the Lard.<\p> <\p>