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Quantum Harmonic Oscillator. The state of a quantum system at t = 0 is initially described by the following wave function Y(x,0) = 3 4 √/30 4/0 (x) + 4 + √30/₁(x) +. /1/4/2₂ (x), where ₁(x), 4₁(x), and ₂ (x) are the first three wave functions of the quantum harmonic oscillator with corresponding energies En = (n +) ħw, where n = 0,1,2. (a) Suppose the energy of this system is measured at t = 0. What are the possible energy outcomes and their corresponding probabilities? (b) Calculate the expectation value of energy at t = 0 (c) What is the wave function (x, t) of this system at a later time t? (d) Calculate the expectation value of energy for any nonzero time t. Compare your result with part (b).

Quantum Harmonic Oscillator. The state of a quantum system at t = 0 is initially described by the following wave function Y(x,0) = 3 4 √/30 4/0 (x) + 4 + √30/₁(x) +. /1/4/2₂ (x), where ₁(x), 4₁(x), and ₂ (x) are the first three wave functions of the quantum harmonic oscillator with corresponding energies En = (n +) ħw, where n = 0,1,2. (a) Suppose the energy of this system is measured at t = 0. What are the possible energy outcomes and their corresponding probabilities? (b) Calculate the expectation value of energy at t = 0 (c) What is the wave function (x, t) of this system at a later time t? (d) Calculate the expectation value of energy for any nonzero time t. Compare your result with part (b).

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Solve the following problems. Write your solutions clearly and in detail in a short bond paper or yellow paper. Quantum Harmonic Oscillator. The state of a quantum system at t = 0 is initially described by the following wave function 4 4(x,0) = √3/30 %0 (x) + 0₁ (x) + √30 ¥₁(x) + √/ 42(x), where ₁(x), 4₁(x), and ₂ (x) are the first three wave functions of the quantum harmonic oscillator with corresponding energies En = (n + 2) ħw, where n = 0, 1, 2. (a) Suppose the energy of this system is measured at t = 0. What are the possible energy outcomes and their corresponding probabilities? (b) Calculate the expectation value of energy at t = 0 (c) What is the wave function (x, t) of this system at a later time t? (d) Calculate the expectation value of energy for any nonzero time t. Compare your result with part (b).
Transcribed Image Text:Solve the following problems. Write your solutions clearly and in detail in a short bond paper or yellow paper. Quantum Harmonic Oscillator. The state of a quantum system at t = 0 is initially described by the following wave function 4 4(x,0) = √3/30 %0 (x) + 0₁ (x) + √30 ¥₁(x) + √/ 42(x), where ₁(x), 4₁(x), and ₂ (x) are the first three wave functions of the quantum harmonic oscillator with corresponding energies En = (n + 2) ħw, where n = 0, 1, 2. (a) Suppose the energy of this system is measured at t = 0. What are the possible energy outcomes and their corresponding probabilities? (b) Calculate the expectation value of energy at t = 0 (c) What is the wave function (x, t) of this system at a later time t? (d) Calculate the expectation value of energy for any nonzero time t. Compare your result with part (b).
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