Hermitian operator represented as a unitary operator

In summary, the equation for U states that the operator defined from it exists and is unitary. Homework Equations state that U=exp(iK) and that M#=M. The Attempt at a Solution states that given a Hermitian operator K, define U as above and then one must prove that exp(iK)*(exp(iK))# = I. However, it is not clear how to do this using only the information given.
  • #1

nomadreid

Gold Member
1,662
203

Homework Statement


I know that any unitary operator U can be realized in terms of some Hermitian operator K (see equation in #2), and it seems to me that it should also be true that, starting from any Hermitian operator K, the operator defined from that equation exists and is unitary.

Homework Equations


U=exp(iK)
Letting M# be the adjoint of M (I can't find a dagger), and * be multiplication,
unitary: M#*M = M*M# = I (Identity operator)
Hermitian:M#=M

The Attempt at a Solution


Given a Hermitian operator K, define U as above, and then one must prove that exp(iK)*(exp(iK))# = I. That is, that (exp(iK))# =exp(-iK)
Obviously I must use the fact that K=K# to do so, but this only makes the problem to show that exp(-iK#) = (exp(iK))# , which doesn't seem to get me very far. The fact that (M#)-1=(M-1)# also seems tempting to apply somewhere, but I do not see where. Oh, of course there is also the possibility that my conjecture is wrong.

Side note: I presume this should go in some Homework rubrik, because although this is not a Homework problem, it seems like something that would be given for homework in an algebra course. Therefore this should probably go under a rubrik for Algebra, but there was no Algebra listing in the Homework menu.
 
Physics news on Phys.org
  • #2
" ... exp(-iK#) = (exp(iK))# ..."

Can you show that? Can you do some operations on exp(iK) that will allow you to bring the # operator into the ()? It's hard to know how much of a hint is considered acceptable in a homework forum, but how about Taylor? So, given that K# = K, what is ( i K * K )#?
 
  • Like
Likes nomadreid
  • #3
Thanks, DEvens. Let me see if I see where your hints are leading. First, since if K is Hermitian so is iK, so
(iK2)#=(iK*K)#=(K#(iK)#)=-i(K#)2.
Similarly
(iK3)#=(iK2*K)#=(iK2)#*K#=
=(-i(K#)2)K#=-i(K#)3
and so forth, so
expanding (exp(iK))# via Taylor, we get (∑ (iK)n/n!)# = (∑ (-iK#)n/n! = exp(-iK#)
Is this correct?
PS I found the dagger M
 

1. What is a Hermitian operator?

A Hermitian operator is a mathematical operator that is equal to its own conjugate transpose. In other words, the operator is symmetric about the main diagonal when written in matrix form.

2. How is a Hermitian operator represented as a unitary operator?

A Hermitian operator can be represented as a unitary operator by finding its eigenvalues and eigenvectors. The eigenvectors can be used to construct a unitary matrix, which, when multiplied by the eigenvalues, gives the original Hermitian operator.

3. What is the significance of a Hermitian operator being represented as a unitary operator?

A Hermitian operator represented as a unitary operator is significant because it means that the operator is both self-adjoint and unitary. This property is important in quantum mechanics, where unitary operators represent the evolution of a quantum system and Hermitian operators represent observables.

4. Can a non-Hermitian operator be represented as a unitary operator?

No, a non-Hermitian operator cannot be represented as a unitary operator. This is because a non-Hermitian operator is not self-adjoint, and therefore cannot be represented by a unitary matrix.

5. How is the unitary operator representation of a Hermitian operator useful in quantum mechanics?

In quantum mechanics, the unitary operator representation of a Hermitian operator is useful because it allows us to easily perform calculations and make predictions about the evolution of a quantum system and the measurement of observables. It also helps us to understand the underlying symmetries and conservation laws of a physical system.

Similar threads

  • Advanced Physics Homework Help
Replies
8
Views
748
  • Advanced Physics Homework Help
Replies
6
Views
8K
Replies
5
Views
1K
  • Advanced Physics Homework Help
Replies
4
Views
3K
Replies
8
Views
2K
  • Advanced Physics Homework Help
Replies
5
Views
5K
  • Advanced Physics Homework Help
Replies
4
Views
3K
  • Advanced Physics Homework Help
Replies
7
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
2K
Back
Top