6. Consider a system of angular momentum l = 1. A basis of its state space...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
6. Consider a system of angular momentum l = 1. A basis of its state space is formed by the three eigenvectors of L2: |+ 1), |0), | – 1), whose eigenvalues are, respectively, +h, 0, and -h, and which satisfy: L+|m) = hv2|m±1) L+|1) = L-| – 1) = 0 This system, which possesses an electric quadrupole moment, is placed in an electric field gradient, so that its Hamiltonian can be written: Wo H = " (L - L) where Lu and L, are the components of L along the two directions Ou and Ov of the xOz plane that form angles of 45° with Ox and Oz; wo is a real constant. a. Write the matrix representing H in the {|+ 1), |0}, | – 1)} basis. Vhat are the stationary states of the system, and what are their energies? (These states are to be written |E1), |E2), |E3), in order of decreasing energies.) b. At time t = 0, the system is in the state: 1 |b(0)) = + 1) –|- 1)] What is the state vector (t)) at time t? At t, Lz is measured; what are the probabilities of the various possible results? c. Calculate the mean values (La)(t), (L,)(t) and (L2)(t) at t. What is the motion performed by the vector (L)? d. At t, a measurement of L is performed. (i) Do times exist when only one result is possible? (ii) Assume that this measurement has yielded the result h2. What is the state of the system immediately after the measurement? Indicate, without calculation, its subsequent evolution. 6. Consider a system of angular momentum l = 1. A basis of its state space is formed by the three eigenvectors of L2: |+ 1), |0), | – 1), whose eigenvalues are, respectively, +h, 0, and -h, and which satisfy: L+|m) = hv2|m±1) L+|1) = L-| – 1) = 0 This system, which possesses an electric quadrupole moment, is placed in an electric field gradient, so that its Hamiltonian can be written: Wo H = " (L - L) where Lu and L, are the components of L along the two directions Ou and Ov of the xOz plane that form angles of 45° with Ox and Oz; wo is a real constant. a. Write the matrix representing H in the {|+ 1), |0}, | – 1)} basis. Vhat are the stationary states of the system, and what are their energies? (These states are to be written |E1), |E2), |E3), in order of decreasing energies.) b. At time t = 0, the system is in the state: 1 |b(0)) = + 1) –|- 1)] What is the state vector (t)) at time t? At t, Lz is measured; what are the probabilities of the various possible results? c. Calculate the mean values (La)(t), (L,)(t) and (L2)(t) at t. What is the motion performed by the vector (L)? d. At t, a measurement of L is performed. (i) Do times exist when only one result is possible? (ii) Assume that this measurement has yielded the result h2. What is the state of the system immediately after the measurement? Indicate, without calculation, its subsequent evolution.
Expert Answer:
Related Book For
Probability & Statistics for Engineers & Scientists
ISBN: 978-0130415295
7th Edition
Authors: Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying
Posted Date:
Students also viewed these physics questions
-
Consider a system of linear equations with augmented matrix A and coefficient matrix C. In each case either prove the statement or give an example showing that it is false. (a) If there is more than...
-
Consider a system of components in which there are five independent components, each of which possesses an operational probability of 0.92. The system does have a redundancy built in such that it...
-
Consider a system of distinguishable particles having only three no degenerate energy levels separated by an energy which is equal to the value of kTat 25.0 K. Calculate (a) The ratio of populations...
-
A total weighted score of in an External Factor Evaluation (EFE) Matrix indicates that an organization is responding in an outstanding way to existing opportunities and threats in its industry. Oa....
-
Update the time series of AMEX and SP500 returns in Section 5.2. Compute their autocorrelation functions and analyze how different or similar they are to those in Figure 5.6.
-
On December 31, 2016 and 2017, Deutsche Telekom AG had net trade receivables in the amount of 9,179 million and 9,553 million, respectively. The following proportion of the receivables was past due...
-
A plant contains 2 centrifugal compressors of carbon steel and \(500-\mathrm{kW}\) rating with explosion-proof electric motors 1 jaw crusher at \(10 \mathrm{~kg} / \mathrm{s}\) capacity 3...
-
Prior to the 2012 Super Bowl, a Boston-area retailer ordered 50,000 T- shirts that read: New England Patriots2012 Super Bowl Champs. The company paid $ 11.75 for each of the custom T- shirts....
-
Princess Inc. is considering a capital project with an investment of $400,000 for cash flows projected for 6 years in the amount of $70,000 with a salvage value of $40,000. Cost of capital is 10% and...
-
The idealized cross-section of a two-cell thin-walled wing box is shown below with the data in Table 22.7. If the wing box supports a load of 44,500 N acting along web 25, calculate the shear flow...
-
Use the sketch of f below along with the fact that g(x) = f(t) dt to answer the questions which follow. (Hint: BEFORE parts c through e, you will need to differentiate both sides) 2-19 (1.4) (2.1)...
-
A single-tank liquid-level system with inflow rate \(q_{i}\) as its input and liquid level \(h\) as its output is modeled as \(R A \dot{h}+g h=R q_{i}(t), h(0)=0\), where \(R, A, g=\) const. If the...
-
The mathematical model of a dynamic system is described by \[4 \ddot{x}+4 \dot{x}+5 x=\frac{10}{3} u_{r}(t), \quad x(0)=0, \quad \dot{x}(0)=\frac{2}{3}\] where \(u_{r}(t)\) is the unit ramp. Plot the...
-
Find the state vector via the formal-solution approach. \(\dot{\mathbf{x}}=\left[\begin{array}{cc}5 & 1 \\ -4 & 1\end{array} ight] \mathbf{x}+\left[\begin{array}{c}1 \\ -1\end{array} ight] u, \quad...
-
Find the frequency response of the given system. \(2 \ddot{x}+5 \dot{x}+8 x=25 \sin 3 t\)
-
The Bode plot of a first-order system indicates that the corner frequency is \(0.35 \mathrm{rad} / \mathrm{sec}\) and high-frequency (at \(\omega=100 \mathrm{rad} / \mathrm{sec}\) ) logarithmic...
-
4. Consider the following functions, where I and J denote two subsets of the set R of real numbers. f: R R x 1/x f(): IJ x f(x) (a) What is the domain of definition of f? (b) Let y be an element of...
-
A company has the following incomplete production budget data for the first quarter: In the previous December, ending inventory was 200 units, which was the minimum required, at 10% of projected...
-
A random sample of 400 voters in a certain city are asked if they favor an additional 4% gasoline sales tax to provide badly needed revenues for street repairs. If more than 220 but fewer than 260...
-
The following data represent the length of life, in seconds, of 50 fruit flies subject to a new spray in a controlled laboratory experiment:(a) Construct a double-stem-and-leaf plot for the life span...
-
For Exercise 12.8, construct a 90% confidence interval for the mean compressive strength when the concentration is x = 19.5 and a quadratic model is used
-
Give a description for three foundation concepts in stability model.
-
Describe three main steps to identifying EBTs.
-
(T/F) EBTs should have a direct physical representation.
Study smarter with the SolutionInn App