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Problem 1 [12] Consider an arbitrary state |psi :| in the Hilbert space of some quantum system. Let A and B be two hermitian observables (operators) on the Hilbert space and denote the expectation values of these observables in the state |psi :| by (:A:)=(:psi |A|psi :) and (:B:)=(:psi |B|psi :). Set U=A-(:A:), V=B-(:B:) and define the state |phi :|, with lambda real. 1.1 Show that (:phi |(phi :)|)=(Delta A)^(2)+lambda ^(2)(Delta B)^(2)+ilambda (:[A,B]:) where (Delta A)^(2)=(:psi |A^(2)|psi :)-(:A:)^(2), (Delta B)^(2)=(:psi |B^(2)|psi :)-(:B:)^(2) and (:[A,B]:)=(:psi |[A,B]|psi :). 1.2 Find the value of lambda that minimizes (:phi |(phi :)|). Using (:phi |(phi :)|)>=0AAlambda , derive the generalized uncertainty relation (Delta A)^(2)(Delta B)^(2)>=(1)/(4)(:i[A,B]:)^(2). 1.3 Discuss the implications of the generalized uncertainty relations in terms of the simultaneous measurement of two non-commuting observables (operators). Use position and momentum as an example to demonstrate your conclusion.

Name: problem 1 12 consider an arbitrary state psi in the hilbert space of some quantum system let a and b be two hermitian observables operators on the hilbert space and denote the expectation va 93146
Uploaded: 2023-02-18T06:05:16-08:00
Duration: 1 min 35 s
Channel: Sri K
Description: problem 1 12 consider an arbitrary state psi in the hilbert space of some quantum system let a and b be two hermitian observables operators on the hilbert space and denote the expectation va 93146

          Problem 1 [12]
Consider an arbitrary state |psi :| in the Hilbert space of some quantum system. Let A and B be two hermitian observables (operators) on the Hilbert space and denote the expectation values of these observables in the state |psi :| by (:A:)=(:psi |A|psi :) and (:B:)=(:psi |B|psi :). Set U=A-(:A:), V=B-(:B:) and define the state |phi :|, with lambda  real.
1.1
Show that
(:phi |(phi :)|)=(Delta A)^(2)+lambda ^(2)(Delta B)^(2)+ilambda (:[A,B]:)
where (Delta A)^(2)=(:psi |A^(2)|psi :)-(:A:)^(2), (Delta B)^(2)=(:psi |B^(2)|psi :)-(:B:)^(2) and (:[A,B]:)=(:psi |[A,B]|psi :).
1.2
Find the value of lambda  that minimizes (:phi |(phi :)|). Using (:phi |(phi :)|)>=0AAlambda , derive the generalized uncertainty relation
(Delta A)^(2)(Delta B)^(2)>=(1)/(4)(:i[A,B]:)^(2).
1.3
Discuss the implications of the generalized uncertainty relations in terms of the simultaneous measurement of two non-commuting observables (operators). Use position and momentum as an example to demonstrate your conclusion.

problem 1 12 consider an arbitrary state psi in the hilbert space of some quantum system let a and b be two hermitian observables operators on the hilbert space and denote the expectation va 93146

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