Which of the following operators is Hermitian? (a) i (b) *...

Which of the following operators is Hermitian? (a) i (b) * (complex conjugate) (c) e^ix (d) x(d/dx)x (e) x d/dx + d/dx x (f) -ix ?/?y (in two dimensions) (g) d^2/dx^2 (in one dimension) (h) -i? ?/?r (in spherical polar coordinates)

Transcript

00:01 Hello everyone so we are doing problem 3 .5 of griffiths here we have to find out the hermitian conjugates of the position operator the imaginary constant i and the derivative operator so the hermitian conjugate of the position operator can be found out using we'll just apply the rule for the inner product here f star xg d xx is equals to since x is real the position of a particle is a real number this will be equal to x f star as x star is equals to x g t x is equals to x fg so as using the rule for the permission conjugate we see that x tagger is actually equals to x so the hermitian conjugate of the position operator is the position operator itself.

01:20 But the next part, the imaginary constant, f, i, g, this equals to integral f star, i, tx, equals to.

01:42 Now, for i to be inside the star bracket, we need it to be equals to minus if.

01:56 Star g because i star is equals to minus i and that minus will cancel with this minus to give i here so this if goes if we want to go from this side to that side we need a minus sign here so minus i f the whole star g t x which is equals to minus i f in a product with g so here so here we have the formation conjugate of i equals to minus i now for the third case the derivative operator f in our product with dg by t x we'll just use the definition integral f star tg by d x now will do the inner product so this becomes product rule for multiplication using the product rule for multiplication we'll get f star g so this derivative just vanished because of the product rule here minus integral minus infinity to infinity now this derivative will go on that side and this will become df by d x star t x now using the rule that at boundaries the both the functions in the hilbert space should vanish so this term will vanish and we will get since there is a minus sign here this will come inside and we'll get this minus of t by tx t f by tx now this minus sign just comes in and this comes minus t by t x of f g so the hermitian conjugate for the derivative operator the whole dagger is actually equals to minus t by t x now for this next part we have we want to find out the hermitian conjugate for the ladder the raising ladder so the raising ladder is given by e plus is equal to one by under root to h cross and omega to minus ip plus m omega x now is the raising operator so it is given by the imaginary constant i p the momentum m omega and x the position operator now p and x have the same hermission conjugate they are equal to their hermission conjugates but i is actually negative of the formation conjugate so we'll use the fact here and we will get a star dagger is equals to 1 by under root 2h cross m omega.

06:33 I will be replaced by minus i.

06:37 So it will become plus i here.

06:40 And p is the same plus m omega and x is also the same...

Question

Which of the following operators is Hermitian? (a) i (b) * (complex conjugate) (c) e^ix (d) x(d/dx)x (e) x d/dx + d/dx x (f) -ix ?/?y (in two dimensions) (g) d^2/dx^2 (in one dimension) (h) -i? ?/?r (in spherical polar coordinates)

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