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We now see that the requirement that the total eigenfunction isantisymmetric with respect to the exchange of the labels of the twoelectrons effectively leads to a coupling between their spin andspatial coordinates. The electrons act as if they move under aforce whose sign depends on the relative orientation of theirspins.
This section is the bonus section for question 1. Consider thehelium atom which is a two- electron atom as we have discussedabove. The ground state is
1s2.
(k) Referring to your list of the s and t total eigenfunctions,write the ground state wave
function for He. Why is this one the only wave function for theground state?
We will now consider the Coulomb electron repulsion in ourtwo-electron atom. You are certainly not expected to do a detailedperturbation theory calculation.

(l)However,justarguingfromyourqualitativeknowledgeofthesandtstates,whichdoyou expect, in general cases other than the ground state, to havethe lower energy and why?

Consider a traveling wave described by the formula

$y_1(x,t) = A \sin(k x - \omega t)$ .

This function might represent the lateral displacement of a string, a local electric field, the position of the surface of a body of water, or any of a number of other physical manifestations of waves.

Part A

Which one of the following statements about the wave described in the problem introduction is correct?

ANSWER:

The wave is traveling in the

direction.
The wave is traveling in the

direction.
The wave is oscillating but not traveling.
The wave is traveling but not oscillating.

Part B

Which of the expressions given is a mathematical expression for a wave of the same amplitude that is traveling in the opposite direction? At time t=0

this new wave should have the same displacement as y_1(x,t)

, the wave described in the problem introduction.

ANSWER:

$A \cos (k x - \omega t)$
$A \cos (k x + \omega t)$
$A \sin (k x - \omega t)$
$A \sin (k x + \omega t)$

The principle of superposition states that if two functions each separately satisfy the wave equation, then the sum (or difference) also satisfies the wave equation. This principle follows from the fact that every term in the wave equation is linear in the amplitude of the wave.

Consider the sum of two waves , where is the wave described in Part A and is the wave described in Part B. These waves have been chosen so that their sum can be written as follows:

$y_{\rm s}(x,t) = y_{\rm e}(x) y_{\rm t}(t)$ .

This form is significant because , called the envelope, depends only on position, and depends only on time. Traditionally, the time function is taken to be a trigonometric function with unit amplitude; that is, the overall amplitude of the wave is written as part of .

Part C
Find and . Keep in mind that should be a trigonometric function of unit amplitude.

erinhyena808

4. 3.5/6.5 points All Submissions Notes Question: SerEssen11.AE.06. .Question part Points Submissions 1 2 3 4 5 6 7 8
.5 .5 .5 1 1 0/1 0/1 0/1
1/15 1/15 1/15 1/15 1/15 5/15 0/15 0/15
Total 3.5/6.5 ... Example 1.6 Cartesian and Polar Coordinates
Goal Understand how to convert from plane rectangular coordinatesto plan polar coordinates and vice versa.

Problem (a) The Cartesian coordinates of a point in the xy-planeare (x, y) = (-3.37, -2.73) m, as shown in Active Figure 1.5 Findthe polar coordinates of this point. (b) Convert (r, ?) = (4.61 m,36.5Â°) to rectangular coordinates.

Strategy Apply the trigonometric functions and their inverses,together with the Pythagorean theorem.
Figure 1.5 Converting from Cartesian coordinates to polarcoordinates.

Solution
(a) Cartesian to Polar
Take the square root of both sides of Equation 1.2 to find theradial coordinate. r =
= 1 m

Use Equation 1.1 for the tangent function to find the angle withthe inverse tangent, adding 180Â° because the angle is actually inthe third quadrant: tan ? = = = 0.810
? = tan-1 (0.810) = 39.0Â° + 180Â°
? = 2Â°

(b) Polar to Cartesian
Use the trigonometric definitions, Equation 1.1. x = r cos? = (4.61m) cos 36.5Â°
x = 3 m

y = r sin? = (4.61 m) sin 36.5Â°
y = 4 m

--------------------------------------------------------------------------------

Remarks When we take up vectors in two dimensions in Chapter 3, wewill routinely use a similar process to find the direction andmagnitude of a given vector from its components, or, conversely, tofind the components from the vector's magnitude anddirection.

Exercise 1.6 Hints: Getting Started | I'm Stuck

(a) Find the polar coordinates corresponding to (x, y) = (3.11,1.28) m.
r = 5 m
? = 6Â°
Your response differs significantly from the correct answer. Reworkyour solution from the beginning and check each stepcarefully.

(b) Find the Cartesian coordinates corresponding to (r, ?) = (4.07m, 52.5Â°)
x = 7 m
y = 8 m

Anything you could do would help, but please explain slowly. Thanksagain. It is for my technical physics 115 class.

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