To show that the change of variables from r 1 , r 2 to u → ≡ 1 2 ( r 1 + r 2 ) , v → ≡ 1 2 ( r 1 − r 2 ) turns the Hamiltonian into two independent three-dimensional harmonic oscillators H = [ − ℏ 2 2 m ∇ u 2 + 1 2 m ω 2 u 2 ] + [ − ℏ 2 2 m ∇ v 2 + 1 2 ( 1 − λ ) m ω 2 v 2 ] .

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Answer 1

Question

Chapter 8, Problem 8.24P

(a)

To determine

To show that the change of variables from r1,r2 to u→≡12(r1+r2) , v→≡12(r1−r2) turns the Hamiltonian into two independent three-dimensional harmonic oscillators H=[−ℏ22m∇u2+12mω2u2]+[−ℏ22m∇v2+12(1−λ)mω2v2] .

(b)

To determine

The exact ground state energy of the system.

(c)

To determine

The ground state energy of the system by applying variational principle and to check the answer with part b.

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Answer 2

Question

Chapter 8, Problem 8.24P

(a)

To determine

To show that the change of variables from r1,r2 to u→≡12(r1+r2) , v→≡12(r1−r2) turns the Hamiltonian into two independent three-dimensional harmonic oscillators H=[−ℏ22m∇u2+12mω2u2]+[−ℏ22m∇v2+12(1−λ)mω2v2] .

(b)

To determine

The exact ground state energy of the system.

(c)

To determine

The ground state energy of the system by applying variational principle and to check the answer with part b.

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Answer 3

Question

Chapter 8, Problem 8.24P

(a)

To determine

To show that the change of variables from r1,r2 to u→≡12(r1+r2) , v→≡12(r1−r2) turns the Hamiltonian into two independent three-dimensional harmonic oscillators H=[−ℏ22m∇u2+12mω2u2]+[−ℏ22m∇v2+12(1−λ)mω2v2] .

(b)

To determine

The exact ground state energy of the system.

(c)

To determine

The ground state energy of the system by applying variational principle and to check the answer with part b.

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Answer 4

Question

Chapter 8, Problem 8.24P

(a)

To determine

To show that the change of variables from r1,r2 to u→≡12(r1+r2) , v→≡12(r1−r2) turns the Hamiltonian into two independent three-dimensional harmonic oscillators H=[−ℏ22m∇u2+12mω2u2]+[−ℏ22m∇v2+12(1−λ)mω2v2] .

(b)

To determine

The exact ground state energy of the system.

(c)

To determine

The ground state energy of the system by applying variational principle and to check the answer with part b.

Expert Solution & Answer

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Answer 5

Chapter 8, Problem 8.24P

(a)

To determine

To show that the change of variables from r1,r2 to u→≡12(r1+r2) , v→≡12(r1−r2) turns the Hamiltonian into two independent three-dimensional harmonic oscillators H=[−ℏ22m∇u2+12mω2u2]+[−ℏ22m∇v2+12(1−λ)mω2v2] .

(b)

To determine

The exact ground state energy of the system.

(c)

To determine

The ground state energy of the system by applying variational principle and to check the answer with part b.

Answer 6

Chapter 8, Problem 8.24P

(a)

To determine

To show that the change of variables from r1,r2 to u→≡12(r1+r2) , v→≡12(r1−r2) turns the Hamiltonian into two independent three-dimensional harmonic oscillators H=[−ℏ22m∇u2+12mω2u2]+[−ℏ22m∇v2+12(1−λ)mω2v2] .

Answer 7

Chapter 8, Problem 8.24P

(a)

To determine

To show that the change of variables from r1,r2 to u→≡12(r1+r2) , v→≡12(r1−r2) turns the Hamiltonian into two independent three-dimensional harmonic oscillators H=[−ℏ22m∇u2+12mω2u2]+[−ℏ22m∇v2+12(1−λ)mω2v2] .

Answer 8

(b)

To determine

The exact ground state energy of the system.

Answer 9

(b)

To determine

The exact ground state energy of the system.

Answer 10

(c)

To determine

The ground state energy of the system by applying variational principle and to check the answer with part b.

Answer 11

(c)

To determine

The ground state energy of the system by applying variational principle and to check the answer with part b.

Answer 12

Expert Solution & Answer

Answer 13

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Answer 14

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Answer 15

Ch. 8.1 - Prob. 8.1P Ch. 8.1 - Prob. 8.2P Ch. 8.1 - Prob. 8.3P Ch. 8.1 - Prob. 8.4P Ch. 8.1 - Prob. 8.5P Ch. 8.1 - Prob. 8.6P Ch. 8.2 - Prob. 8.7P Ch. 8.2 - Prob. 8.8P Ch. 8.3 - Prob. 8.9P Ch. 8.3 - Prob. 8.10P

Solution Summary: The author explains that the change of variables from r_1,

To show that the change of variables from r1,r2 to u→≡12(r1+r2) , v→≡12(r1−r2) turns the Hamiltonian into two independent three-dimensional harmonic oscillators H=[−ℏ22m∇u2+12mω2u2]+[−ℏ22m∇v2+12(1−λ)mω2v2] .

The exact ground state energy of the system.

The ground state energy of the system by applying variational principle and to check the answer with part b.

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Chapter 8 Solutions