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a) Prove that (x) = 0 for the ground state of a harmonic oscillator. b) Prove that (x²) 2Vuk for the ground state of a harmonic oscillator. c) Use the result in (b) to calculate the root-mean-square amplitude of 14N2 in its ground state. Use k = 2260 N m¬1 for 14N2. The mass of 14N = 14.003 amu. d) The value you obtained in (c) is a measure of how much the bond length in this molecule varies (about its equilibrium length) due to vibrational motion. Given that the equilibrium bond length of 14N2 is 109.77 pm, by how much does the bond length vary due to vibrational motion, as a percent of its equilibrium value?

a) Prove that (x) = 0 for the ground state of a harmonic oscillator. b) Prove that (x²) 2Vuk for the ground state of a harmonic oscillator. c) Use the result in (b) to calculate the root-mean-square amplitude of 14N2 in its ground state. Use k = 2260 N m¬1 for 14N2. The mass of 14N = 14.003 amu. d) The value you obtained in (c) is a measure of how much the bond length in this molecule varies (about its equilibrium length) due to vibrational motion. Given that the equilibrium bond length of 14N2 is 109.77 pm, by how much does the bond length vary due to vibrational motion, as a percent of its equilibrium value?

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a) Prove that (x) = 0 for the ground state of a harmonic oscillator. b) Prove that (x²) 2Vuk for the ground state of a harmonic oscillator. c) Use the result in (b) to calculate the root-mean-square amplitude of 14N2 in its ground state. Use k = 2260 N m¬1 for 14N2. The mass of 14N = 14.003 amu. d) The value you obtained in (c) is a measure of how much the bond length in this molecule varies (about its equilibrium length) due to vibrational motion. Given that the equilibrium bond length of 14N2 is 109.77 pm, by how much does the bond length vary due to vibrational motion, as a percent of its equilibrium value?
Transcribed Image Text:a) Prove that (x) = 0 for the ground state of a harmonic oscillator. b) Prove that (x²) 2Vuk for the ground state of a harmonic oscillator. c) Use the result in (b) to calculate the root-mean-square amplitude of 14N2 in its ground state. Use k = 2260 N m¬1 for 14N2. The mass of 14N = 14.003 amu. d) The value you obtained in (c) is a measure of how much the bond length in this molecule varies (about its equilibrium length) due to vibrational motion. Given that the equilibrium bond length of 14N2 is 109.77 pm, by how much does the bond length vary due to vibrational motion, as a percent of its equilibrium value?
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