Question

Consider an observable with continuous spectra. In contrast with the discrete case, what can you say about the eigenfunctions of this observable, the probability of obtaining a given value after a measurement, and the state of a system after a measurement is performed?

Answer 1

;For discrete case, the wave function can be expressed in terms of the observable as:

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where, |\phi_i \rangle are the corresponding eigenstates of the observable, and |C_i|^2 is the probability for that observable to have the corresponding eigenstate.

In the continuous case, this is expressed as :

V = C(k)|ok)dk

So, in this scenario, a specific eigen function of the observable is |\phi_k \rangle which need not to be discrete, for example, the position eigenfunction is .00) = 51.1 - .00) , which depicts a position at 1 - 2 , but, position isn't discrete, it's continuous over the space, so that we integrate it from - infinity to + infinity to get the wave function, similarly for momentum eigenfunction :

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One object can not have always discrete values of allowed momentum, thus, the allowed values are continuous naturally.

For an observable with continuous spectra, one realistic object can't have an exact value of the observable without any uncertainty.

Now, probability for finding the value of the observable within the range k=ki to ka in a continuous spectra is found by :

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As, the spectra is continuous, the object can not have a specific value of the observable, rather there is a little range to be taken to find the probability.

Now, after measurement, for discrete spectra of an observable, the wave function must only posses that corresponding Eigen function, whereas for continuous spectra, wave function must collapse to that corresponding eigen function as a delta function.

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