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

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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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A cylinder with a piston contains 0.153 mol of nitrogen at a pressure of 1.83×105 Pa and a temperature of 290 K. The nitrogen may be treated as an ideal gas. The gas is first compressed isobarically to half its original volume. It then expands adiabatically back to its original volume, and finally it is heated isochorically to its original pressure. Part A Compute the temperature at the beginning of the adiabatic expansion. Express your answer in kelvins. ΕΠΙ ΑΣΦ T₁ = ? K Submit Request Answer Part B Compute the temperature at the end of the adiabatic expansion. Express your answer in kelvins. Π ΑΣΦ T₂ = Submit Request Answer Part C Compute the minimum pressure. Express your answer in pascals. ΕΠΙ ΑΣΦ P = Submit Request Answer ? ? K Pa

Learning Goal: To understand the meaning and the basic applications of pV diagrams for an ideal gas. As you know, the parameters of an ideal gas are described by the equation pV = nRT, where p is the pressure of the gas, V is the volume of the gas, n is the number of moles, R is the universal gas constant, and T is the absolute temperature of the gas. It follows that, for a portion of an ideal gas, pV = constant. Τ One can see that, if the amount of gas remains constant, it is impossible to change just one parameter of the gas: At least one more parameter would also change. For instance, if the pressure of the gas is changed, we can be sure that either the volume or the temperature of the gas (or, maybe, both!) would also change. To explore these changes, it is often convenient to draw a graph showing one parameter as a function of the other. Although there are many choices of axes, the most common one is a plot of pressure as a function of volume: a pV diagram. In this problem, you…

Learning Goal: To understand the meaning and the basic applications of pV diagrams for an ideal gas. As you know, the parameters of an ideal gas are described by the equation pV = nRT, where p is the pressure of the gas, V is the volume of the gas, n is the number of moles, R is the universal gas constant, and T is the absolute temperature of the gas. It follows that, for a portion of an ideal gas, pV = constant. T One can see that, if the amount of gas remains constant, it is impossible to change just one parameter of the gas: At least one more parameter would also change. For instance, if the pressure of the gas is changed, we can be sure that either the volume or the temperature of the gas (or, maybe, both!) would also change. To explore these changes, it is often convenient to draw a graph showing one parameter as a function of the other. Although there are many choices of axes, the most common one is a plot of pressure as a function of volume: a pV diagram. In this problem, you…

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