An individual who experiences powerful feelings of both joy and sadness would score high on what personality dimension? neuroticism affect variability affect intensity affect instability

## An individual who experiences powerful feelings of both joy and sadness would score high on what personality dimension? neuroticism affect variability affect intensity affect instability

An individual who experiences powerful feelings of both joy and … Read More...
1. (20 pts) The linear momentum operator in one dimension is given by: ˆpx = ! i d dx . In class we said that the average momentum for a particle in a box (pib) is 0. Use the formula for the expectation value to verify mathematically that this is true. The pib-wavefunction: 2 ℓ sin nπ ℓ x ! ” # \$ % & 2. (20 pts) Evaluate the following commutators: a. (10 pts) b. (10 pts) 3. (20 pts) A certain one-dimensional quantum mechanical system is described by the Hamiltonian: , q is a constant, and 0 ≤ x ≤ ∞. One of the eigenfunctions is known to be: a. (15 pts) Find the value of N to normalize the function. b. (5 pts) By letting , find the energy eigenvalue. 4. (20 pts) The Schrödinger equation for the particle on a sphere (a.k.a. the Rigid Rotor) is: − !2 2μR2 1 sinθ ∂ ∂θ sinθ ∂ ∂θ # \$ % & ‘ ( + 1 sin2θ ∂2 ∂φ 2 # \$ % & ‘ ( ψ(θ,φ ) = Eψ(θ,φ ) A purported eigenfunction for it is: ψ(θ,φ ) = N sin3θ cos(3φ ) a. (15 pts) Use this wave function to find the energy eigenvalue for the function. (You do NOT have to normalize the function!). b. (5 pts) The eigenvalues for the particle on a sphere are of the form: Eℓ = “2 2μR2 ℓ(ℓ +1) What is the value of ℓ for the wave function used in part a? 5. (20 pts) Using the ortho-normailty of the hydrogenic orbitals and the spin functions, normalize the excited Helium atom represented by the following wave function: ψ = N 1s({ 1)2p(2)+ 2p(1)1s(2)}{α(1)β (2)−β (1)α(2)} ˆ x, ( ˆpx [ + xˆ)] = ˆpx, ( ˆ x)3 !” #\$ = ˆH = − 2 2m d2 dx2 − q2 x ψ(x) = Nxe−α x α = mq2 / 2

## 1. (20 pts) The linear momentum operator in one dimension is given by: ˆpx = ! i d dx . In class we said that the average momentum for a particle in a box (pib) is 0. Use the formula for the expectation value to verify mathematically that this is true. The pib-wavefunction: 2 ℓ sin nπ ℓ x ! ” # \$ % & 2. (20 pts) Evaluate the following commutators: a. (10 pts) b. (10 pts) 3. (20 pts) A certain one-dimensional quantum mechanical system is described by the Hamiltonian: , q is a constant, and 0 ≤ x ≤ ∞. One of the eigenfunctions is known to be: a. (15 pts) Find the value of N to normalize the function. b. (5 pts) By letting , find the energy eigenvalue. 4. (20 pts) The Schrödinger equation for the particle on a sphere (a.k.a. the Rigid Rotor) is: − !2 2μR2 1 sinθ ∂ ∂θ sinθ ∂ ∂θ # \$ % & ‘ ( + 1 sin2θ ∂2 ∂φ 2 # \$ % & ‘ ( ψ(θ,φ ) = Eψ(θ,φ ) A purported eigenfunction for it is: ψ(θ,φ ) = N sin3θ cos(3φ ) a. (15 pts) Use this wave function to find the energy eigenvalue for the function. (You do NOT have to normalize the function!). b. (5 pts) The eigenvalues for the particle on a sphere are of the form: Eℓ = “2 2μR2 ℓ(ℓ +1) What is the value of ℓ for the wave function used in part a? 5. (20 pts) Using the ortho-normailty of the hydrogenic orbitals and the spin functions, normalize the excited Helium atom represented by the following wave function: ψ = N 1s({ 1)2p(2)+ 2p(1)1s(2)}{α(1)β (2)−β (1)α(2)} ˆ x, ( ˆpx [ + xˆ)] = ˆpx, ( ˆ x)3 !” #\$ = ˆH = − 2 2m d2 dx2 − q2 x ψ(x) = Nxe−α x α = mq2 / 2

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Chapter 9 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, April 18, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Momentum and Internal Forces Learning Goal: To understand the concept of total momentum for a system of objects and the effect of the internal forces on the total momentum. We begin by introducing the following terms: System: Any collection of objects, either pointlike or extended. In many momentum-related problems, you have a certain freedom in choosing the objects to be considered as your system. Making a wise choice is often a crucial step in solving the problem. Internal force: Any force interaction between two objects belonging to the chosen system. Let us stress that both interacting objects must belong to the system. External force: Any force interaction between objects at least one of which does not belong to the chosen system; in other words, at least one of the objects is external to the system. Closed system: a system that is not subject to any external forces. Total momentum: The vector sum of the individual momenta of all objects constituting the system. In this problem, you will analyze a system composed of two blocks, 1 and 2, of respective masses and . To simplify the analysis, we will make several assumptions: The blocks can move in only one dimension, namely, 1. along the x axis. 2. The masses of the blocks remain constant. 3. The system is closed. At time , the x components of the velocity and the acceleration of block 1 are denoted by and . Similarly, the x components of the velocity and acceleration of block 2 are denoted by and . In this problem, you will show that the total momentum of the system is not changed by the presence of internal forces. m1 m2 t v1(t) a1 (t) v2 (t) a2 (t)