Unitary and self-adjoint operators

[the_kid]

Homework Statement

Let U and A be two linear maps related by U=e^iA. Show that U is unitary if A is self-adjoint. Give a counterexample to show that U can be unitary if A is not self-adjoint.

Homework Equations

Self-adjoint: A*=A

The Attempt at a Solution

OK, so I had no problem with the first part. It's easy to show that U dagger=U inverse. However, I'm having some trouble coming up with a counter example for the second part. I tried something anti-Hermitian, but that just shows that U is self-adjoint, not that it is unitary.

 

[micromass]

Write out what it means that $UU^*=I=U^*U$. What could go wrong?

 

[the_kid]

Wait. What do you mean "what could go wrong"? I need to show that U can be unitary even if A isn't self-adjoint by way of an example.

 

[micromass]

Wait. What do you mean "what could go wrong"? I need to show that U can be unitary even if A isn't self-adjoint by way of an example.

I didn't phrase that the best way as I could.

So, indeed, you need to find a non-self-adjoint matrix A such that $U=e^{iA}$ is unitary. So, assume that U is unitary, then $UU^*=I=U^*U$. What does that mean for A??

 

[the_kid]

I didn't phrase that the best way as I could.

So, indeed, you need to find a non-self-adjoint matrix A such that $U=e^{iA}$ is unitary. So, assume that U is unitary, then $UU^*=I=U^*U$. What does that mean for A??

$UU^*=(e^{iA})(e^{iA})^+=(e^{iA})^+(e^{iA})=U^*U$

I feel like this is really simple; I wonder why I'm not seeing it.

 

[micromass]

$UU^*=(e^{iA})(e^{iA})^+=(e^{iA})^+(e^{iA})=U^*U$

I feel like this is really simple; I wonder why I'm not seeing it.

OK, so you want $UU^*=I=UU^*$. You indeed say that this is the same as

$$(e^{iA})(e^{iA})^+=I=(e^{iA})^+(e^{iA})$$

Now, do you know what $(e^{iA})^+$ is?? Can you write it in another way??

 

[the_kid]

OK, so you want $UU^*=I=UU^*$. You indeed say that this is the same as

$$(e^{iA})(e^{iA})^+=I=(e^{iA})^+(e^{iA})$$

Now, do you know what $(e^{iA})^+$ is?? Can you write it in another way??

$(e^{iA})^+=e^{-iA^+}$

 

[micromass]

OK, so now you have

$$e^{iA}e^{-iA^+}=I=e^{-iA^+}e^{iA}$$

Now, what happens if you use $e^{A+B}=e^A e^B$?? (I know the rule isn't always valid for matrices, but let's assume $AA^+=A^+A$, in that case the rule is valid).

 

[the_kid]

OK, so now you have

$$e^{iA}e^{-iA^+}=I=e^{-iA^+}e^{iA}$$

Now, what happens if you use $e^{A+B}=e^A e^B$?? (I know the rule isn't always valid for matrices, but let's assume $AA^+=A^+A$, in that case the rule is valid).

You get $$e^{i(-A^++A)}=I=e^{i(A-A^+)}$$.

 

[micromass]

OK, so what did we find now? We found out that if A isn't self-adjoint, then there are two ways that $e^{iA}$ could still be unitary.

Either, $B=A+A^+$ is a matrix such that $e^{iB}=I$.
Or $AA^+\neq A^+A$.

One of those two things must hold for A.

Let's focus on the second one, can you find a matrix such that $AA^+\neq A^+A$.

 

[the_kid]

OK, so what did we find now? We found out that if A isn't self-adjoint, then there are two ways that $e^{iA}$ could still be unitary.

Either, $B=A+A^+$ is a matrix such that $e^{iB}=I$.
Or $AA^+\neq A^+A$.

One of those two things must hold for A.

Let's focus on the second one, can you find a matrix such that $AA^+\neq A^+A$.

I think I can come up with a matrix, but can't we just treat these as operators, too? How about the ladder operator in QM. It is non-Hermitian; i.e. $AA^+\neq A^+A$.

 

[micromass]

I think I can come up with a matrix, but can't we just treat these as operators, too? How about the ladder operator in QM. It is non-Hermitian; i.e. $AA^+\neq A^+A$.

The condition $AA^+=A^+A$ is known as "normal".
But anyway, what do you mean with the ladder operator??
Matrices are fine examples of operators. Matrices are exactly the operators on a finite dimensional space. So any counterexamples you find as matrices will be a counterexample of operators.

 

[the_kid]

The condition $AA^+=A^+A$ is known as "normal".
But anyway, what do you mean with the ladder operator??
Matrices are fine examples of operators. Matrices are exactly the operators on a finite dimensional space. So any counterexamples you find as matrices will be a counterexample of operators.

http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator#Ladder_operator_method

I am terrible at Latex, so I'm just linking to Wikipedia. I think this makes sense, right?

 

[micromass]

http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator#Ladder_operator_method

I am terrible at Latex, so I'm just linking to Wikipedia. I think this makes sense, right?

Oh, those. I call them shift operators

Anyway, maybe those ladder operators work, but for that you got to calculate $e^{iA}$ and that seems a little tedious.
The easiest calculations are, in my opinion, with matrices.

 

[the_kid]

Oh, those. I call them shift operators

Anyway, maybe those ladder operators work, but for that you got to calculate $e^{iA}$ and that seems a little tedious.
The easiest calculations are, in my opinion, with matrices.

OK, how about let A=({2,3}, {-2,1})? I think that works.

 

[micromass]

OK, how about let A=({2,3}, {-2,1})? I think that works.

OK, but the calculations are a bit annoying. Can't we find a matrix whose exponential can easily be calculated? For example, try a nilpotent matrix.

 

[the_kid]

OK, but the calculations are a bit annoying. Can't we find a matrix whose exponential can easily be calculated? For example, try a nilpotent matrix.

OK, how about ({0,1}, {0,0})?

 

[micromass]

OK, how about ({0,1}, {0,0})?

Yes, try that one! What do you get?

 

[the_kid]

Yes, try that one! What do you get?

You get A dagger does not commute with A, as we wanted. What else do I need to show?

 

[the_kid]

What do I need to show from there?