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Intro to Quantum Mechanics problem:
. In a harmonic oscillator a normalized "coherent" state ya(x) is...
5. Coherent States (Answer only question 5 for part a, b, and c) A coherent state is an Eigenstat...
5. Coherent States (Answer only question 5 for part a, b, and
c)
A coherent state is an Eigenstate of annihilation / lowering
operator
c) Baker- Campbell- Hausdorff Formula [Hint: Define the functions fa-eaA+8), ģ(A)-eä eABe-12 č. Note that these functions are equal at -0, and show that they satisfy the same differential equation: df/di (A+ B)f and dg/da (A+B)g Therefore, the functions are themselves equal for all λ.] A useful application of BCH formula is given in problem 5...
5. A particle in the harmonic oscillator potential has the initial wave function Psi(x, 0) = A[\psi_{0}(x) + \psi_{1}(x)] for some constant A. Here to and ₁ are the normalized ground state and the first excited state wavefunctions of the harmonic oscill
5. A particle in the harmonic oscillator potential has the initial wave function Psi(x, 0) = A[\psi_{0}(x) + \psi_{1}(x)] for some constant A. Here to and ₁ are the normalized ground state and the first excited state wavefunctions of the harmonic oscillator, respectively. (a) Normalize (r, 0). (b) Find the wavefunction (r, t) at a later time t and hence evaluate (x, t) 2. Leave your answers involving expressions in to and ₁. c) sing the following normalized expression of...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian 12 12 m2 H = -2m d. 2+ 2 Here m and w are the mass and frequency, respectively. Consider a time-dependent wave function of the form <(x,t) = C'exp (-a(x – 9(t)+ ik(t)z +io(t)), where a and C are positive constants, and g(t), k(t), and o(t) are real functions of time t. 1. Express C in terms of a. [2 marks] 2. By...
QUANTUM MECHANICS Problem 4 Consider a one-dimensional charged harmonic oscillator. Let the coordinate be, charge be...
QUANTUM MECHANICS
Problem 4 Consider a one-dimensional charged harmonic oscillator. Let the coordinate be, charge be q, mass be m, and the frequency of the oscillator be u. (a) 79 rat t =-oo, the oscillator is in the ground state 10). A uniform electric field E along x axis is applied betweentoo andtoo with the time dependence of E being given by E(t) ー(t/ア Neglect the induced magnetic field. Find the probability that the oscillator goes to the nth excited...
Problem 5. (30 points) Consider a Harmonic oscillator with H that H=(ata + 1 / 2)ho,...
Problem 5. (30 points) Consider a Harmonic oscillator with H that H=(ata + 1 / 2)ho, where a=dma)X + i (a) (4 points) Show P, and a x 2h 2h 2moh P. Show also 2moh that [a, a]-l. (b) (6 points) Starting from the commuters la, HJand la', A), where H-H(h) show that the eigenvalues of Hare e,=(n+1/2) for n-0, 1,2, Show also that alm)-nln-l), and a l). (( points) Find the normalized ground state wavefunction by projecting alo)-0 on...
2. [10 points] For a Simple Harmonic Oscillator: a) Draw ψ(x) and ψ2(x) for the u,...
2. [10 points] For a Simple Harmonic Oscillator: a) Draw ψ(x) and ψ2(x) for the u, and v-6 states. Make sure to include as much detail as possible. b) Show that 2(3) is normalized. 3. [5] A certain non-halogen diatomic molecule was found to have a force constant of 99 N/m and an observed vibrational frequency of 162.2 cm1 Determine the identity of the unknown diatomic 4. [10 points) a) What is the difference between commuting and non-commuting operators? What...
1 From Wavefunction to Bra-Ket In bra-ket notation, a state y(x) is written as a ket:...
1 From Wavefunction to Bra-Ket In bra-ket notation, a state y(x) is written as a ket: 14) + (2). The inner product between two states 41(2), 42(2) is written as a bra-ket: (441\42) = |dx (z)* #2(a). If a state is a complex linear combination |V) = a1 (41) + a2 (42), then its corresponding bra is (V= a1 (01| +a(42 In this problem, we will use the simple harmonic oscillator as a concrete example. The energy eigenstates of the...
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force pr...
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force problem (as discussed in TZD Many physical systems involve a particle that moves under the influence of a central force; that is, a force that always points exactly toward, or away from, a force center O. In classical mechanics a famous example of a central force is the force of the sun on a planet. In atomic physics the most obvious example is the hydrogen atom,...
5. Coherent States (Answer only question 5 for part a, b, and
c)
A coherent state is an Eigenstate of annihilation / lowering
operator
c) Baker- Campbell- Hausdorff Formula [Hint: Define the functions fa-eaA+8), ģ(A)-eä eABe-12 č. Note that these functions are equal at -0, and show that they satisfy the same differential equation: df/di (A+ B)f and dg/da (A+B)g Therefore, the functions are themselves equal for all λ.] A useful application of BCH formula is given in problem 5...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian 12 12 m2 H = -2m d. 2+ 2 Here m and w are the mass and frequency, respectively. Consider a time-dependent wave function of the form <(x,t) = C'exp (-a(x – 9(t)+ ik(t)z +io(t)), where a and C are positive constants, and g(t), k(t), and o(t) are real functions of time t. 1. Express C in terms of a. [2 marks] 2. By...
QUANTUM MECHANICS
Problem 4 Consider a one-dimensional charged harmonic oscillator. Let the coordinate be, charge be q, mass be m, and the frequency of the oscillator be u. (a) 79 rat t =-oo, the oscillator is in the ground state 10). A uniform electric field E along x axis is applied betweentoo andtoo with the time dependence of E being given by E(t) ー(t/ア Neglect the induced magnetic field. Find the probability that the oscillator goes to the nth excited...
Problem 5. (30 points) Consider a Harmonic oscillator with H that H=(ata + 1 / 2)ho, where a=dma)X + i (a) (4 points) Show P, and a x 2h 2h 2moh P. Show also 2moh that [a, a]-l. (b) (6 points) Starting from the commuters la, HJand la', A), where H-H(h) show that the eigenvalues of Hare e,=(n+1/2) for n-0, 1,2, Show also that alm)-nln-l), and a l). (( points) Find the normalized ground state wavefunction by projecting alo)-0 on...
2. [10 points] For a Simple Harmonic Oscillator: a) Draw ψ(x) and ψ2(x) for the u, and v-6 states. Make sure to include as much detail as possible. b) Show that 2(3) is normalized. 3. [5] A certain non-halogen diatomic molecule was found to have a force constant of 99 N/m and an observed vibrational frequency of 162.2 cm1 Determine the identity of the unknown diatomic 4. [10 points) a) What is the difference between commuting and non-commuting operators? What...
1 From Wavefunction to Bra-Ket In bra-ket notation, a state y(x) is written as a ket: 14) + (2). The inner product between two states 41(2), 42(2) is written as a bra-ket: (441\42) = |dx (z)* #2(a). If a state is a complex linear combination |V) = a1 (41) + a2 (42), then its corresponding bra is (V= a1 (01| +a(42 In this problem, we will use the simple harmonic oscillator as a concrete example. The energy eigenstates of the...
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force problem (as discussed in TZD Many physical systems involve a particle that moves under the influence of a central force; that is, a force that always points exactly toward, or away from, a force center O. In classical mechanics a famous example of a central force is the force of the sun on a planet. In atomic physics the most obvious example is the hydrogen atom,...
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