2 Quant. questions (expanding an abitrary state, transition amplitude?)

[ausdreamer]

Homework Statement

1) The position operator x has a continuous spectrum of engenvalues {x'} with corresponding eigenkets {|x'>}. Expand this state in terms of the position eigenstates.

2) Suppose the transition amplitudes c_a',b' = <a'|b'> are known where {|b'>} are the eigenstates of a Hermition operator. Write cos(A) in terms of {|b'>}.

Homework Equations

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The Attempt at a Solution

1) Is this correct? Let the arbitrary state be |x>, then:

|x> = integral(dx' * |x'><x'| * |x>) ?

2) I have no idea how one would write cos(A) in terms of {|a'>}, let alone in terms of {|b'>}??

Any help appreciated!

 

[grey_earl]

Normally you would normalize the position eigenstates such that <x'|x> = δ(x-x'), so that you get |x> = ∫ |x'> <x'|x> dx' = ∫ |x'> δ(x-x') dx' = |x>, which is a tautology. I don't quite get the reasoning behind the first question, expand a position eigenstate in terms of position eigenstates? Maybe it was meant momentum eigenstates.

For the second part, use that cos(A) = ½ [ exp(i A) + exp(-i A) ], which is defined by the series expansion of the exponential. Then you can use that cos(A) = cos(A) Σ_{a'} |a'><a'| = Σ_{a'} ( cos(A) |a'> ) <a'| and so on.

 

[lanedance]

For the first i would ask what state though? it doesn't specifically say position eigenstate, is a state referenced somewhere? ro could you assume soem arbitrary state |\phi&gt; [\itex]

 

[ausdreamer]

So we can write:

cos(A) = sum_a' [(cos(A) |a'>) <a'|] = sum_a' [sum_b' [|b'><b'| (cos(A) |a'>) <a'|] ] = sum_a' [sum_b' [cos(A) <b'|a'> <a'|b'>] ] = sum_c [ c^2 * cos(A)],

since <b'|a'> = <a'|b'>* = <a'|b'> = c (given in the question), since A,B hermitian? OR should I do it like this?

cos(A) = sum_a' [(cos(A) |a'>) <a'|] = sum_a' [sum_b' [|b'><b'| (cos(A) |a'>) <a'|] ] = sum_a' [sum_b' [<b'|cos(A)|b'> |a'> <a'|] ] = sum_b' [<b'|cos(A)|b'>]

Both methods and results seem correct, but if I write it in terms of c, then I don't end up writing cos(A) in terms of {|b'>} as it asks...



As for the first question, I forgot to write it all out: "The position operator x has a continuous spectrum of engenvalues {x'} with corresponding eigenkets {|x'>}. The momentum operator p has a continuous set of eigenvalues {p'} with corresponding eigenkets {|p'>}. An ensemble of particles is in an arbitrary state |alpha>. Write out this state in terms of the position eigenstates."

Is the answer to this still just: |alpha> = integral [dx' delta(x-x')* |x'>], like you wrote grey_earl?

Thanks very much for your replies so far, it has been a great help to my study :)

 

[lanedance]

no, that is only if the state is a position eigenstate |x>, replace it with alpha

 

[ausdreamer]

Ah I see, thanks for that lancedance :) There's more parts to this question that I might post here in case anyone sees this in the next 12 hours or so before my exam I'm studying for :)...

(Continuing on from the arbitrary state |alpha>...) Suppose we make a measurement that selects only particles in the range x'-A/2 to x'+A/2, where A is some small real number. What is the normalized state of these particles immediately after this measurement?

So the new state after the measurement would be (call it beta)... |beta> = integral_lower bounds x'-A/2, upper bounds x+A/2 [dx' |x'><x'|alpha>] = integral[dx' |x'> dirac(alpha-x')]

But have I found the "normalized state"? what do i have to do if i haven't?



Thanks so much for all help so far :)

 

[grey_earl]

So we can write:

cos(A) = sum_a' [(cos(A) |a'>) <a'|] = sum_a' [sum_b' [|b'><b'| (cos(A) |a'>) <a'|] ] = sum_a' [sum_b' [cos(A) <b'|a'> <a'|b'>] ] = sum_c [ c^2 * cos(A)],

Before introducing the sum over b', you should apply cos(A) on the state |a'>, that's why I put it in parentesis. Coming from the definition, you get cos(A) |a'> = Σ_{n=0}^∞ (-1)^n/(2 n)! A^(2n) |a'> = Σ_{n=0}^∞ (-1)^n/(2 n)! (a')^(2n) |a'> = cos(a') |a'>. Then write |a'> = Σ_(b') |b'> <b'|a'> and an analog expression for <a'| to get the final result. Note that you cannot change |b'><a'| to <a'|b'>, since the first is an operator, but the second is a number.

As for the first question, I forgot to write it all out: "The position operator x has a continuous spectrum of engenvalues {x'} with corresponding eigenkets {|x'>}. The momentum operator p has a continuous set of eigenvalues {p'} with corresponding eigenkets {|p'>}. An ensemble of particles is in an arbitrary state |alpha>. Write out this state in terms of the position eigenstates."

Is the answer to this still just: |alpha> = integral [dx' delta(x-x')* |x'>], like you wrote grey_earl?

No, since <x'|α> is not equal to a δ distribution. You just have |α> = ∫ |x'><x'| dx' |α> = ∫ <x'|α> |x'> dx' = ∫ α(x') |x'> dx'.

Thanks very much for your replies so far, it has been a great help to my study :)

:) Any time again!