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Q1: Write down the general expression for the expectation value of an operator Ô for aquantum mechanical particle described by a wave function V(x, t). Briefly describe theconnection between operators, wave functions, expectation values and experimental mea-surements.See Answer
Q2:2. (30 points) A quantum mechanical particle is in an eigenstate ) of Î² with eigenvalue 2ħ²: At a particular moment, the particle also is in an eigenstate of the x component of the angular momentum, Î, with eigenvalue 0. In other words, . Express this eigenstate of I, as a normalized superposition of the familiar eigenstates, |lm), of Î² and Îz, where Î¿ is the z component of the angular momentum.See Answer
Q3:The spin operator in an arbitrary direction can be written as (0,0) = sin cos o + y sin sin + ₂ cos 0, I where the Pauli spin matrices are given by 0 (13), 1. Find the eigenvectors and eigenvalues for the operator ô(0, 6). ÔT x = Oy = 0 (² i). i 0 Oz = 0 0 -1See Answer
Q4:2. Choose = 0 and find for an entangled state of the form 0 0 |I) >- (B),B),-),B)) 1/12 = 2 1 0 1 0 the probability of detecting particle 1 in spin-up with respect to an angle 0₁ and at the same time particle 2 in spin-up with respect to an angle 02.See Answer
Q5:4. A light wave is specified as following (in SI unites), Find: Ẽ =(−68 +3√5ŷ)(10¹) expil (√5x+2y)π×10² −9.42×10¹³t] . (1) The direction of electric field. (2) The scalar value of the amplitude of the electric field. (3) The direction of propagation. (4) The wave number and wavelength. (See Answer
Q6:Problem 3: Adiabatic Limit Re-consider the exponential switch perturbation, but now we will focus on extremely small values of the switch rate, a. This will lead us to an important theorem in Quantum Mechanics called the Adiabatic Theorem. The Adiabatic Theorem states that a system will stay in its evolving eigenstate provided the Hamiltonian is changed sufficiently slowly. Test this claim using the parameters of [1c]: x = 0.1 w₁ and a = 0.001 w₁₁. You already have the numerical and perturbative solutions to this transition. Now generate a third plot by directly calculating the occupation of the lowest eigenstate of H = Ho + V in the basis of Ho. This should allow you to calculate the occupation probability of the excited state of Ho as a function of time. Plot this probability along with your numerical and perturbative results, and comment on what you find. Aside: This is not the same notion of "adiabatic" as in thermal systems. Vaik ⓇSee Answer
Q7:X 1. Light travels in a piece of glass with its electric field E₂ = E cos710¹5 (t- 0.65c angular frequency, wavelength, and index of refraction of glass, and its intensity. -). Find itsSee Answer
Q8:4. The full width at half maximum of an atomic absorption line at 589 nm is 100 MHz. A beam of light passes through a gas with an atomic density of 10¹7/m³. Calculate: (a) the peak absorption coefficient due to this absorption line. You can assume that the index of refraction is close to 1 in for this dilute gas; (b) the frequency at which the resonant contribution to the refractive index is at a maximum; (c) the peak value of this resonant contribution to the index of refraction.See Answer
Q9:[b] Plot the spectral density, D[], for a fixed value of & and several different values of T. (Put these all on the same plot, carefully labeling everything.) These plots should make clear that a finite lifetime, 1, implies a finite line width--i.e. a spectrum that peaks at & but is spread out in a Gaussian-like distribution around this energy as shown below./nKDmax A SE Dmax 2 Verify that the width of the distribution, is equal to 1/7 at the half-height of the peak. This is typically inter- preted as the range of energies that you might expect to measure in an experiment. It implies that τ δε = 1. (4) This is called the Lifetime Broadening Relation, and it should call to mind the time-energy uncertainty relation. Explain what the LB Relation says about the certainty with which you can know the excited state energy as a function of the lifetime of the excited state. Look at the two extremes, zero lifetime and infinite lifetime, to help elucidate the physics.See Answer
Q10:#1 Calculate the radius of the first (n=1) Bohr orbit for the hydrogen atom. (10 points) Mass of an electron, m = 9.1e-³¹kgSee Answer
Q11:#3 a) Calculate the energy in Joules of a green laser (wavelength of 530 nm) in a vacuum. (5 points) b) Convert the answer to energy in electron volts (eV) (5 points)See Answer
Q12:#2 Calculate the energy of the first (n=1) Bohr orbit for the hydrogen atom in eV (10 points)See Answer
Q13:#4 Calculate the wavelength of an electron in the ground state for a Hydrogen Atom (10 points) Velocity = 2.2X106. and mass = 9.1X10-³¹ kg Would you expect this electron with a diameter of 1.0X10-10 m to exhibit wave behavior? Why?See Answer
Q14:#9 Give the quantum numbers (n=1,2,3,etc) shell, and the subshell designation (s,p,d) for each of the following elements A: Neon (Ne) (5 points) B: Arsenic (As) (5 points) Assignment #2 C: Gallium (Ga) (5 points) C: Copper (Cr) (5 points) D. Indium (In) (5 points) 3 ELEC 311 Fall 2023See Answer
Q15:#7 calculate the first three energy levels (n=1,2,3) in eV for a 3Angstrom quantum well with infinite walls (15 points) Mass of an electron, m = 9.1e-31 kgSee Answer
Q16:D Question 1 The position of the maximum probability density is the position where [Select] ✓ [Select] 1/4 aol. 4 4*4 *r density is greatest at r = [Select] 4 pts is greatest, ne ernately, lity e ing the solving ectron, ity [in units of/nU Question 2 D Question 1 The position of the maximum probability density is the position where [Select] this you can get from inspection of the [Select] ✓ [Select] aol. 2 pts volume-weighted, squared, wavefunction 4 pts is greatest, wavefunction (or normalized wavefunction) normalized wavefunction . Alternately, lity e ing the solving ectron, ity units of/nD Question 1 The position of the maximum probability density is the position where [Select] this you can get from inspection of the [Select] 0 0.135 4 pts you could find the maximum probability density by taking the derivative of the probability density function, and setting the resulting equation equal to zero, and solving for the roots. In the case of the 2s electron, ity ✓ [Select] 1 is greatest. 2 Question 2 Alternately, units of 2 ptsSee Answer
Q17:D Question 2 2 pts The most probable distance from the nucleus for the 2s electron is r = [Select] [in units of aol./nD Question 2 The most probable distance from the nucleus for the 2s electron is r = ✓ [Select] 5.23 5 4.33 2 1 0 0.76 2 pts 6 units of 2 pts ance, or n 2s inits ofSee Answer
Q18:D Question 3 2 pts The average distance, mean distance, or expected distance for the hydrogen 2s electron is r = [Select] aol. [in units of/nQuestion 3 ✓ [Select] 6 CO 5.23 LO 5 4.33 4 3 0.76 N 0 Upload 2 pts ance, or n 2s inits of 6 pts helpful.See Answer
Q19:aol. D Question 4 6 pts Upload your written work for these subproblems. Note: graphs are very helpful.See Answer
Q20:Draw an energy-level diagram showing the lowest four levels of singly ionized helium. Show all possible transitions from the levels and label each transition with its wavelength. 08:45 ✓/See Answer

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