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1. Consider a harmonic oscillator sitting in the ground state with a given spring constant ko m w...
2. Now consider a particle in the ground state of the harmonic oscillator. ok gives the...
2. Now consider a particle in the ground state of the harmonic oscillator. ok gives the wave function for the ground state, but not the value of the constant A. Determine what it has to be if the ground state is normalized. (b) Suppose a classical particle has an energy equal to the ground state energy E. This particle will, of course, oscillate back and forth as though it were attached to a spring. What would its turning points be?...
Consider a harmonic oscillator with Hamiltonian given by ?=(p^2/2m)+(1/2)X^2 = (a+)(a-)+(1/2) The current system state is...
Consider a harmonic oscillator with Hamiltonian given by ?=(p^2/2m)+(1/2)X^2 = (a+)(a-)+(1/2) The current system state is the superposition of the lowest and next-to-lowest energy eigenstates that gives the most negative possible value for the average position, use raising and lowering operators to derive the average momentum for this state. then, simplify using ħ = ? = 1
A H2 molecule can be approximated by a simple harmonic oscillator with spring constant 1000 N/m....
A H2 molecule can be approximated by a simple harmonic oscillator with spring constant 1000 N/m. Note: you must use the reduced mass µ H = 1 2mH for this kind of problem. (a) Find the ground state energy in eV. (b) Find all possible wavelengths of photons emitted as the molecule decays from the third excited state eventually to the ground state.
Consider the dimensionless harmonic oscillator Hamiltonian, (where m = h̄ = 1). Consider the orthogonal wave...
Consider the dimensionless harmonic oscillator Hamiltonian,
(where m = h̄ = 1).
Consider the orthogonal wave functions
and
, which are eigenfunctions of H with eigenvalues 1/2 and 5/2,
respectively.
with p=_ïda 2 2 We were unable to transcribe this imageY;(r) = (1-2x2)e-r2/2 (a) Let фо(x-AgVo(x) and φ2(x) = A2V2(x) and suppose that φ。(x) and φ2(x) are normalized. Find the constants Ao and A2. (b) Suppose that, at timet0, the state of the oscillator is given by Find the constant...
For the simple harmonic oscillator ground state, because()0 and (p) 0 (expectation values of z and...
For the simple harmonic oscillator ground state, because()0 and (p) 0 (expectation values of z and p are - Ar and Using this fact, you can estimate the ground state energy. Follow steps below for this calculation. a. For SHO potential V(z)--mw,2, write down the total energy of the ground state in terms of ΔΖ2 and p2. and constant parameters that characterize the SHO (m and w) total energy- Format Hint: Write(z*) as Δ2. not (Δε)2. The system considers two...
Discuss Concept Problem 1 1. A mass M is horizontally attached to a spring with a spring constant k. Let's consider two different inputs: a) A sudden step force Fo in the downward direction. Anal...
Discuss Concept Problem 1 1. A mass M is horizontally attached to a spring with a spring constant k. Let's consider two different inputs: a) A sudden step force Fo in the downward direction. Analyze the resulting motion, like amplitude and mean value. Now include a damper: How can you find an engineering characterization of a "Damper"? What will the final displacement be, whern including the damper? (System at rest) . b) An oscillatory force F A sin(wt) will be...
Quantum Mechanics Problem 1. (25) Consider an infinite potential well with the following shape: 0 a/4...
Quantum Mechanics Problem
1. (25) Consider an infinite potential well with the following shape: 0 a/4 3al4 a h2 where 4 Using the ground state wavefunction of the original infinite potential well as a trial function, 2πχ trial = 1-sin- find the approximation of the ground state energy for this system with the variational method. (Note, this question is simplified by considering the two components of the Hamiltonian, and V, on their own) b) If we had used the 1st...
1 Sert9 7Pte A ight spring with spring constant ,30 x 10 N/m hangs from an...
1 Sert9 7Pte A ight spring with spring constant ,30 x 10 N/m hangs from an elevated support. From Rs lower end hangs a second Sght spring, which has spering constant 1.95x10 N/m A 1.50-kg object hangs at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs m (b) Find the effective spring constant of the pair of springs as a system. We describe these springs as being in...
6. The energy levels of a harmonic oscillator with angular frequency w are given by 2...
6. The energy levels of a harmonic oscillator with angular frequency w are given by 2 (a) Suppose that a system of N almost independent oscillators has total energy E^Nhw 2 Mhw. Show that the number of states with exactly this energy equals the number of ways of distributing M identical objects among N compartments and that this number 1S MI(N 1) Hint: Consider the number of distinct arrangements of a set of M objects and N -1 partitions (b)...
1. (30pt) LC Circuit and Simple Harmonic Oscillator (From $23.12 RLC Series AC Circuits) Let us...
1. (30pt) LC Circuit and Simple Harmonic Oscillator (From $23.12 RLC Series AC Circuits) Let us first consider a point mass m > 0 with a spring k> 0 (see Figure 23.52). This system is sometimes called a simple harmonic oscillator. The equation of motion (EMI) is given by ma= -kr (1) where the acceleration a is given by the second derivative of the coordinate r with respect to time t, namely dr(t) (2) dt de(t) (6) at) (3) dt...
2. Now consider a particle in the ground state of the harmonic oscillator. ok gives the wave function for the ground state, but not the value of the constant A. Determine what it has to be if the ground state is normalized. (b) Suppose a classical particle has an energy equal to the ground state energy E. This particle will, of course, oscillate back and forth as though it were attached to a spring. What would its turning points be?...
Consider the dimensionless harmonic oscillator Hamiltonian,
(where m = h̄ = 1).
Consider the orthogonal wave functions
and
, which are eigenfunctions of H with eigenvalues 1/2 and 5/2,
respectively.
with p=_ïda 2 2 We were unable to transcribe this imageY;(r) = (1-2x2)e-r2/2 (a) Let фо(x-AgVo(x) and φ2(x) = A2V2(x) and suppose that φ。(x) and φ2(x) are normalized. Find the constants Ao and A2. (b) Suppose that, at timet0, the state of the oscillator is given by Find the constant...
For the simple harmonic oscillator ground state, because()0 and (p) 0 (expectation values of z and p are - Ar and Using this fact, you can estimate the ground state energy. Follow steps below for this calculation. a. For SHO potential V(z)--mw,2, write down the total energy of the ground state in terms of ΔΖ2 and p2. and constant parameters that characterize the SHO (m and w) total energy- Format Hint: Write(z*) as Δ2. not (Δε)2. The system considers two...
Discuss Concept Problem 1 1. A mass M is horizontally attached to a spring with a spring constant k. Let's consider two different inputs: a) A sudden step force Fo in the downward direction. Analyze the resulting motion, like amplitude and mean value. Now include a damper: How can you find an engineering characterization of a "Damper"? What will the final displacement be, whern including the damper? (System at rest) . b) An oscillatory force F A sin(wt) will be...
Quantum Mechanics Problem
1. (25) Consider an infinite potential well with the following shape: 0 a/4 3al4 a h2 where 4 Using the ground state wavefunction of the original infinite potential well as a trial function, 2πχ trial = 1-sin- find the approximation of the ground state energy for this system with the variational method. (Note, this question is simplified by considering the two components of the Hamiltonian, and V, on their own) b) If we had used the 1st...
1 Sert9 7Pte A ight spring with spring constant ,30 x 10 N/m hangs from an elevated support. From Rs lower end hangs a second Sght spring, which has spering constant 1.95x10 N/m A 1.50-kg object hangs at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs m (b) Find the effective spring constant of the pair of springs as a system. We describe these springs as being in...
6. The energy levels of a harmonic oscillator with angular frequency w are given by 2 (a) Suppose that a system of N almost independent oscillators has total energy E^Nhw 2 Mhw. Show that the number of states with exactly this energy equals the number of ways of distributing M identical objects among N compartments and that this number 1S MI(N 1) Hint: Consider the number of distinct arrangements of a set of M objects and N -1 partitions (b)...
1. (30pt) LC Circuit and Simple Harmonic Oscillator (From $23.12 RLC Series AC Circuits) Let us first consider a point mass m > 0 with a spring k> 0 (see Figure 23.52). This system is sometimes called a simple harmonic oscillator. The equation of motion (EMI) is given by ma= -kr (1) where the acceleration a is given by the second derivative of the coordinate r with respect to time t, namely dr(t) (2) dt de(t) (6) at) (3) dt...
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