How to Find the Expectation Value of Total Energy?

[Chronos000]

Homework Statement

given a wavefuntion $$\Psi$$ = (1/sqrt50) (3$$\mu$$₁ + 5$$\mu$$₂ - 4$$\mu$$₃)

what is the expectation value of the total energy?


My thoughts were to calculate <$$\Psi$$|$$\hat{}H$$|$$\Psi$$>

but the previous part to the question asks for the probability of each outcome(which I know how to find). So is there a way to do this using the probabilities?

 

[kloptok]

For a discrete probability distribution $$f(x_i)$$ [x (and thus f) takes only discrete values], the expectation value of a quantity $$x$$ is

$$<x>= \sum^N_i x_i f(x_i)$$,

where x takes on values $$x_1, x_2, \ldots , x_N$$

For a continuous probability distribution $$g(x)$$ [where g and x are continous], the expectation value of x is the limit of the sum, namely the integral

$$<x>= \int^{x_{max}}_{x_{min}} x g(x)$$.

So if you know the probability distribution (which it seems like you do) the rest is basic maths.

 

[Chronos000]

so are you saying that the answer is just E_T = 9/50 E₁ + 25/50 E₂ + 16/50 E₃ ?

 

[ideasrule]

Yes.