Let H be the Hamiltonian operator of a physical system. Denote by | , > the...

 

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Let H be the Hamiltonian operator of a physical system. Denote by | , > the eigenvectors of H, with eigenvalues E: H|₁>= En|9n> n a. For an arbitrary operator A, prove the relation: <on | [A, H]|n) = 0. b. Consider a one-dimensional problem, where the physical system is a particle of mass m and of potential energy V(X). In this case, H is written: 1 H = p² + V(X) 2m x. In terms of P, X and V(X), find the commutators: [H, P], [H, X] and [H, XP]. B. Show that the matrix element ( 9₁ | P | Q₁ > (which we shall interpret in chapter III as the mean value of the momentum in the state | ₁ >) is zero. n p² Pn 2m dv kinetic energy in the state | , >) and < 9₁ | X an>. Since the mean value of the potential energy in the state | 9₁ ) is < 9₂ | V(x) | q₁), how is it related to the mean value of the kinetic energy when: = y. Establish a relation between E₁² V(X) = V₁ X² (λ = 2, 4, 6...; V > 0)? <@nl. | ₁ ) (the mean value of the Let H be the Hamiltonian operator of a physical system. Denote by | , > the eigenvectors of H, with eigenvalues E: H|₁>= En|9n> n a. For an arbitrary operator A, prove the relation: <on | [A, H]|n) = 0. b. Consider a one-dimensional problem, where the physical system is a particle of mass m and of potential energy V(X). In this case, H is written: 1 H = p² + V(X) 2m x. In terms of P, X and V(X), find the commutators: [H, P], [H, X] and [H, XP]. B. Show that the matrix element ( 9₁ | P | Q₁ > (which we shall interpret in chapter III as the mean value of the momentum in the state | ₁ >) is zero. n p² Pn 2m dv kinetic energy in the state | , >) and < 9₁ | X an>. Since the mean value of the potential energy in the state | 9₁ ) is < 9₂ | V(x) | q₁), how is it related to the mean value of the kinetic energy when: = y. Establish a relation between E₁² V(X) = V₁ X² (λ = 2, 4, 6...; V > 0)? <@nl. | ₁ ) (the mean value of the

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