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The general uncertainty relation between two observables A and B.

[tex](\Delta A)^2(\Detla B)^2 \geq -{1\over 4}<[A, B]>^2[/tex]

I have to prove the above relation using the definition of expection values etc.

The reference I use (Liboff) have this relation given as an exercise. But Gasiorowicz's book has given some useful hints on this relation. But I couldn't go all the way to prove this result.

Here is my attempt at it.

<Takes a deep breath>

[tex](\Delta A)^2 = <(A - <A>)^2>[/tex]

[tex](\Delta B)^2 = <(B - <B>)^2>[/tex]

Let U = A - <A> & V = B - <B> and consider [itex]\phi = U\psi + i\lamba V \psi[/itex]

The uncertainties in A & B would be correlated only if the two operators do not commute.

Let A & B be Herimtian so U & V will also be Hermitian.

[tex]I(\lambda) = \int dx \phi^* \phi \geq 0[/tex]

[tex]I(\lambda) = \int dx(U\psi + i\lamba V \psi)^*(U\psi + i\lamba V \psi) = \int dx \psi^* [U^2 + \lambda^2 V^2 +i\lambda[U,V]\psi[/tex]

Using the defintion of [itex]\Delta A & \Delta B[/itex]

[tex]I(\lambda) = \left((\Delta A)^2 + \lambda^2 (\Delta B)^2 + i\lambda<[A, B]>\right) \geq 0[/tex]

I have to get rid of [itex]\lambda[/itex] to get my result. Anyone has good idea here??

BTW, Happy New Year to all the hardworking homework helpers in PF. Keep up the great work!

[tex](\Delta A)^2(\Detla B)^2 \geq -{1\over 4}<[A, B]>^2[/tex]

I have to prove the above relation using the definition of expection values etc.

The reference I use (Liboff) have this relation given as an exercise. But Gasiorowicz's book has given some useful hints on this relation. But I couldn't go all the way to prove this result.

Here is my attempt at it.

<Takes a deep breath>

[tex](\Delta A)^2 = <(A - <A>)^2>[/tex]

[tex](\Delta B)^2 = <(B - <B>)^2>[/tex]

Let U = A - <A> & V = B - <B> and consider [itex]\phi = U\psi + i\lamba V \psi[/itex]

The uncertainties in A & B would be correlated only if the two operators do not commute.

Let A & B be Herimtian so U & V will also be Hermitian.

[tex]I(\lambda) = \int dx \phi^* \phi \geq 0[/tex]

[tex]I(\lambda) = \int dx(U\psi + i\lamba V \psi)^*(U\psi + i\lamba V \psi) = \int dx \psi^* [U^2 + \lambda^2 V^2 +i\lambda[U,V]\psi[/tex]

Using the defintion of [itex]\Delta A & \Delta B[/itex]

[tex]I(\lambda) = \left((\Delta A)^2 + \lambda^2 (\Delta B)^2 + i\lambda<[A, B]>\right) \geq 0[/tex]

I have to get rid of [itex]\lambda[/itex] to get my result. Anyone has good idea here??

BTW, Happy New Year to all the hardworking homework helpers in PF. Keep up the great work!

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