Problem 2. (30pts) Calculate the T matrix in real space representation T(z;r, r') using the Lipmann-Schwinger equation T(z) = V + V GO(P)V +VGo()VGo()V +... = V [1 +Go(z)V + (Go(z)V)2 + ... ] (3) for the delta potential V(r) = 98(r). (4) (Hint1: the real space representation for T matrix is T(z;r, r') = (r|T(2)|r'). (5) Hint2: matrix/operator products in real space should be written with integration AB + (r| AB 1"} = du" ("] 4 [r")<"Bly"). = dr" (6) Hint3: for the delta potential, we have (r| V |r") = V(r)8(r – p"). (7) Hint4: (r|Go(2)|r') = m exp(iv2mz|r – p'). 271 1r – p!! (8)
Answers
The representation for T matrix in real space for the given delta V(r) = g δ(r).
To calculate the T matrix in real space representation for the given delta potential V(r) = g δ(r), we will use the Lipmann-Schwinger equation and the provided hints.
The Lipmann-Schwinger equation for the T matrix is given by:
T(z) = V + V G₀(z) + V G₀(z) V G₀(z) + ...
Let's start by expanding the equation:
T(z) = V + V G₀(z) + V G₀(z) V G₀(z) + ...
= V [1 + G₀(z) + (V G₀(z))^2 + ...]
Now, we need to express each term in the real space representation using the given hints.
The real space representation for the T matrix is given by:
T(z; r, r') = (r | T(z) | r')
Using Hint 2, we can write the matrix/operator products in real space as integrals:
T(z; r, r') = (r | V | r') + ∫ dr'' (r | V G₀(z) | r'') (r'' | V | r')
For the delta potential, we have:
(r | V | r'') = V(r) δ(r - r')
Substituting this into the equation, we get:
T(z; r, r') = V(r) δ(r - r') + ∫ dr'' (r | V G₀(z) | r'') (r'' | V | r')
Using Hint 3 and Hint 4, we have:
T(z; r, r') = g δ(r - r') + ∫ dr'' (r | V G₀(z) | r'') V(r'') δ(r'' - r')
Simplifying the equation, we get:
T(z; r, r') = g δ(r - r') + ∫ dr'' g (exp(i √(2mz)) / (4π |r - r''|)) δ(r'' - r')
Now, we can integrate over the delta function δ(r'' - r'):
T(z; r, r') = g δ(r - r') + g (exp(i √(2mz)) / (4π |r - r'|))
Therefore, the expression for the T matrix in real space representation is:
T(z; r, r') = g δ(r - r') + g (exp(i √(2mz)) / (4π |r - r'|))
Complete Question:
Calculate the T matrix in real space representation, T(z;r, r'), using the Lipmann-Schwinger equation T(z) = V + V G₀(z) + V G₀(z) V G₀(z) + ... = \(V[1+G_0(z) + (V G_0(z))^2+ ...]\) for the delta potential V(r) = gδ(r).
Hint 1: The real space representation for the T matrix is T(z;r, r') = (r|T(z)|r').
Hint 2: Matrix/operator products in real space should be written with integration. For example, AB + (r|AB|r') = ∫ dr'' (r|A|r'') (r''|B|r').
Hint 3: For the delta potential, we have (r|V|r") = V(r)δ(r – r").
Hint 4: (r|G₀(z)|r') = (-m/2π) (exp(i√(2mz)|r-r'|) / (|r-r'|)).
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Related Questions
Question 4 of 10
Which of the following could be the ratio between the lengths of the two legs
of a 30-60-90 triangle?
Check all that apply.
□A. √2:√2
B. 15
□ C. √√√√5
□ D. 12
DE √3:3
OF. √2:√5
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SUBMIT
Answers
The ratios that could be the lengths of the two legs in a 30-60-90 triangle are √3:3 (option E) and 12√3 (option D).
In a 30-60-90 triangle, the angles are in the ratio of 1:2:3. The sides of this triangle are in a specific ratio that is consistent for all triangles with these angles. Let's analyze the given options to determine which ones could be the ratio between the lengths of the two legs.
A. √2:√2
The ratio √2:√2 simplifies to 1:1, which is not the correct ratio for a 30-60-90 triangle. Therefore, option A is not applicable.
B. 15
This is a specific value and not a ratio. Therefore, option B is not applicable.
C. √√√√5
The expression √√√√5 is not a well-defined mathematical operation. Therefore, option C is not applicable.
D. 12√3
This is the correct ratio for a 30-60-90 triangle. The ratio of the longer leg to the shorter leg is √3:1, which simplifies to √3:3. Therefore, option D is applicable.
E. √3:3
This is the correct ratio for a 30-60-90 triangle. The ratio of the longer leg to the shorter leg is √3:1, which is equivalent to √3:3. Therefore, option E is applicable.
F. √2:√5
This ratio does not match the ratio of the sides in a 30-60-90 triangle. Therefore, option F is not applicable. So, the correct option is D. 1 √2.
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In a right triangle two of the angle measurements are equal find the value of X the number of degrees in each of the equal angle measurements
Answers
The unknown angles of the right angle triangle is 45 degrees.
How to find the angle of a right triangle?A right angle triangle is a triangle with one angle equals to 90 degrees. The sum of angles in a triangle is 180 degrees.
Therefore, the right triangle has two angle measurements that are equal to each other. The third angle is 90 degrees.
Therefore, let's find the angles measure x, the two other equal angles.
Hence,
x + x + 90 = 180
2x + 90 = 180
subtract 90 from both sides of the equation
2x + 90 = 180
2x + 90 - 90 = 180 - 90
2x = 90
divide both sides by 2
x = 90 / 2
x = 45 degrees
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3√1728=?
Don't report me
Jordan please keep fighting for me.
Answers
Answer:
The answer is 12
Step-by-step explanation:
trust
PLEASE MARK BRAINLIEST!!!!!!!!!!!!!!!!!i need itan irs auditor randomly selects 3 tax returns from59returns of which7contain errors. what is the probabilitythat she selects none of those containing errors?
Answers
Therefore , the probability that she selects none of those containing errors is 0.6798.
What does probability really mean?Calculating how likely or "possible" it is for an event to occur is the subject of probability. Words like "definitely," "impossible," or "likely" may be used to communicate the likelihood that a particular event will occur. Mathematics always expresses probabilities as fractions, decimals, or percentages, with values ranging from 0 to 1.
Here,
Number of ways of selecting 3 returns out of 59 is C(59,3).
Out of 59 returns, 59-7 = 52 returns has no error so number of ways of selecting 3 returns out fo 52 is C(52,3).
The probability that she selects none of those containing errors is
P( No errors )=\(\frac{C(52,3)}{C(59,3)}\)=0.6798
Therefore , the probability that she selects none of those containing errors is 0.6798.
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Using a single letter, list the angles from largest to smallest.
Answers
We can guess that the widest angles are opposite to the largest sides.
Then, we can order the angels from largest to smallest as:
M, Q and O.
Determine whether these are valid arguments.a) If x is a positive real number, then x² is a positive real number. Therefore, if a² is positive, where a is a real number, then a is a positive real number.b) If x² ≠ 0, where x is a real number, then x ≠ 0. Let a be a real number with a² ≠ 0; then a ≠ 0.
Answers
The given argument is " If x is a positive real number, then x² is a positive real number." is invalid. Because if a² is positive, it is not always true that a is a positive real number. The given argument is " If x² ≠ 0, where x is a real number, then x ≠ 0. Let a be a real number with a² ≠ 0; then a ≠ 0." is valid argument. Because if a is real number so a² is also a real number.
If x is a positive real number, then x² is a positive real number. Therefore, if a² is positive, where a is a real number, then a is a positive real number. This is an invalid argument. The premise states that if x is a positive real number, then x² is a positive real number, which is true. However, the conclusion that if a² is positive, then a is a positive real number does not necessarily follow from the premise. For example, if a is -2, then a² is positive, but a is not a positive real number.
If x² ≠ 0, where x is a real number, then x ≠ 0. Let a be a real number with a² ≠ 0; then a ≠ 0. This is a valid argument. The premise states that if x² ≠ 0, then x ≠ 0, which is true since the only way for x² to be equal to zero is if x is equal to zero. The conclusion that if a² ≠ 0, then a ≠ 0 follows logically from the premise, since a is a real number and the same logic applies to a² as to x².
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sfsdfsdsdsdgfdgfdgdfignore this!
Answers
Answer:
what lol
Step-by-step explanation:
What is the difference between a histogram and a cumulative histogram?
Answers
Answer: The answer is provided below
Step-by-step explanation:
A histogram is a diagram which consist of rectangles whereby the area is proportional to frequency of a variable and the width is equal to class interval. A histogram is a commonly used graph that is used to show frequency distributions.
The cumulative histogram is a histogram whereby the vertical axis doesn't gives only the counts for a single bin, but gives the counts for that bin and all the bins for the maller values of a response variable.
Cumulative histograms are similar to normal histograms, but the main difference is that they graph cumulative frequencies unlike histograms that graph just frequencies.
The equation of line a is y=12x−1, and it passes through point (6,2).
Line b is perpendicular to line a, and it passes through point (−6,2).
A: What is the slope of line b?
B: What is the y-intercept of line b?
Answers
Answer:
A). slope = -1/12 y-intercept = 3/2
Step-by-step explanation:
y = -1/12x + b
2 = -1/12(-6) + b
2 = 1/2 + b
3/2 = b
y = -1/12x + 3/2
Solve this equation for me pls pls pls pls pls pls pls
Answers
Answer:
8.12857143
Step-by-step explanation:
Caleb took 24 photos at the zoo. 3/8 of his photos are of giraffes. How many of his photos are
of giraffes?
Answers
Answer:
9
Step-by-step explanation:
divide 24 by 8 and multiply what you got by 3
24/8=3
3*3=9
Answer:
9
Step-by-step explanation:
\( \frac{3}{8} \times \frac{24}{1} \)
\( \frac{3 \times 24}{8 \times 1} \)
\( \frac{72}{8} \)
\(72 \div 8\)
\( = 9\)
what is the volume of a regular octahedron inside a cube with the length of its sides measuring 4 centimeters
Answers
Answer:
We conclude that the volume of a regular octahedron will be:
\(\:\:V\:\approx \:30.17\) cmStep-by-step explanation:
Given
The length of one of the side or edge of a regular octahedron = a = 4 cm
To determine
The volume of a regular octahedron = V
Using the formula to determine the volume of a regular octahedron
\(V\:=\:\frac{\sqrt{2}}{3}a^3\)
substituting a = 4 in the volume equation
\(V=\frac{\sqrt{2}\cdot \:4^3}{3}\)
\(V=\frac{64\sqrt{2}}{3}\) ∵ 4³ = 64
\(\:\:V\:\approx \:30.17\) cm
Therefore, we conclude that the volume of a regular octahedron will be:
\(\:\:V\:\approx \:30.17\) cmDetermine if the following triangle is acute, right, or obtuse.
Answers
Answer: It is an acute angle
Step-by-step explanation:
The angles are 45.35, 39.52, 95.13
What is the perimeter of the given polygon
Answers
Use the solution method from this example to find a basis for the given subspace. S = span {[1 -1 0 2], [3 -5 4 8], [0 1 -2 -1]} Give the dimension of the basis. v
Answers
Answer:
Step-by-step explanation:
The dimension of the basis is {[1 0 0 2], [-1 1 0 0]}.
To find a basis for the subspace S = span {[1 -1 0 2], [3 -5 4 8], [0 1 -2 -1]}, we can use the same method as in the example. First, we put the vectors in a matrix and row-reduce it:
[1 -1 0 2]
[3 -5 4 8]
[0 1 -2 -1]
R2 - 3R1 -> R2
R3 -> R3 + 2R1
[1 -1 0 2]
[0 -2 4 2]
[0 1 -2 -1]
-1/2R2 -> R2
[1 -1 0 2]
[0 1 -2 -1]
[0 1 -2 -1]
R3 - R2 -> R3
[1 -1 0 2]
[0 1 -2 -1]
[0 0 0 0]
We can see that the last row is all zeros, so we have only two pivots and one free variable. This means that the dimension of the subspace S is 2. To find a basis, we can write the pivots as linear combinations of the original vectors:
[1 -1 0 2] = [1 0 0 2] + [-1 1 0 0]
[0 1 -2 -1] = [0 1 -2 -1]
Therefore, a basis for S is {[1 0 0 2], [-1 1 0 0]}.
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Tiara simplifies an equation and makes the inference that it has no solutions, which equation would cause her to make that inference ?
Answers
Answer:
C) \(3=7\)
Step-by-step explanation:
Let's try the equations one by one.
\(x=4\) has the solution, well, \(x=4\), so that's not the answer.
\(0=0\) is always true, so that's not the answer.
\(3=7\) is always not true, so that must be the answer.
So, the answer is C) \(3=7\)
I am trying to find the value of x, y, and z.
Answers
Answer:
\(\begin{gathered} x=\sqrt[]{35}=5.62 \\ y=\sqrt[]{60}=7.75 \\ z=\sqrt[]{84}=9.17 \end{gathered}\)Explanation: We need to find x y z, three missing sides:
From the given triangle, we can form three equations, using the Pythagorean theorem as:
\(\begin{gathered} (1)\rightarrow5^2+x^2=y^2 \\ (2)\rightarrow7^2+x^2=z^2 \\ (3)\rightarrow z^2+y^2=(7+5)^2=12^2 \\ \end{gathered}\)Solution by substitution:
Substituting (1) in (3) gives:
\(\begin{gathered} z^2+5^2+x^2=12^2 \\ z^2+25+x^2=144 \\ z^2+x^2=144-25=119 \\ z^2+x^2=119\rightarrow(4) \end{gathered}\)We have reached equation (4), solving for z in equation (4), and then substituting it into equation (2) gives:
\(\begin{gathered} 7^2+x^2=119-x^2\rightarrow2x^2=119-49=70 \\ \therefore\rightarrow \\ x=\sqrt[]{\frac{70}{2}}=\sqrt[]{35}=5.916 \end{gathered}\)Plugging this x into equation (4) gives:
\(\begin{gathered} z^2+(\sqrt[]{35})^2=119\rightarrow z^2=119-35=84 \\ \therefore\rightarrow \\ z=\sqrt[]{84}=9.165 \\ \\ \end{gathered}\)Now that we have x and z, we plug x it into equation (1) and we get:
\(\begin{gathered} 5^2+(\sqrt[]{35})^2=y^2 \\ \therefore\rightarrow \\ y^2=25+35=60 \\ y=\sqrt[]{60}=7.75 \end{gathered}\)x y z respectively are:
\(\begin{gathered} x=\sqrt[]{35}=5.62 \\ y=\sqrt[]{60}=7.75 \\ z=\sqrt[]{84}=9.17 \end{gathered}\)What is the measure of Zx in the diagram shown below?
A
23°
B
33°
С
113°
157°
Answers
Answer:
Since that is a right angle it is going to equal 90 degrees. Since you know that the first angle is 67 degrees subtract 67 from 90 and get 23 degrees.
Step-by-step explanation:
A tree is 4. feet tall. It is expected to grow by 1 feet per year.
At this growth rate, how many years will it take for the tree to have a height greater than 40 feet? Enter the answer in the box.
years
Answers
Answer:
37 years
Step-by-step explanation:
Answer:
36 years
Step-by-step explanation:
it would take 36 years to reach the height of 40 feet. I know this because it is already 4 feet and needs to grow 36 feet and if it is growing at a rate of 1 foot per year then it would take 36 years
A jury pool consists of 12 angry people and 8 chill people. How many 12 member juries:Can be made?Consist of 7 chill and 5 angry people?Consist of at least 10 angry people?
Answers
Given that the number of angry people=12 and the number of chill peoples=8.
The total number of peoples=The number of angry people + the number of chill people.
The total number of people in juries=12+8=20
Let A be an event that made 12 members of juries consists of 7 chill and 5 angry people.
The probability of A =P(A)
\(P(A)=\frac{8C_7+12C_5}{20C_{12}}\)\(=(\frac{8!}{7!(8-7)!}+\frac{12!}{5!(12-5)!})\div\frac{20!}{12!(20-12)!}\)\(=(\frac{8\times7!}{7!}+\frac{12\times11\times10\times9\times8\times7!}{5\times4\times3\times2\times(7)!})\div\frac{20\times19\times18\times17\times16\times15\times13\times12!}{12!\times8\times7\times6\times5\times4\times3\times2}\)\(=800\div8997.857=0.0889\)\(P(A)=0.09\)Hence only one 12 member jury pool can be made that consist of 7 chill and 5 angry people.
Hence the answer is 1.
If you take the opposite of the product of 8 and -2, will the answer be less than -5, between -5 and 5 and 10, or greater than 10?
Answers
Answer: Greater than 10.
8x-2=-16 the inverse (opposite) of -16 is 16. 16 is greater than 10.
For circular motion on a circle of radius r, linear speed equals angular speed divided by r. (T/F)
Answers
For circular motion on a circle of radius r, linear speed equals angular speed divided by r. This statement is false.
What is angular speed?
The definition of angular speed is the rate at which angular displacement changes, and it is written as follows -
ω = θ/t
where θ is the angular displacement, t is the time and ω is the angular speed.
The statement says that for circular motion on a circle of radius r, linear speed equals angular speed divided by r.
Consider that an object moves around a circle of radius r at a constant speed v.
If s is the distance travelled in time t around the circle then linear speed v is defined as v = s/t.
Also if θ is the angle swept out by this object in time t then the angular speed is defined as ω = θ/t.
Thus, there is some relationship between linear speed and angular speed -
Linear speed = v = s/t
= rθ/t = r(θ/t) = rω
where is ω measured in radians per unit time and for a circle of radius r, a central angle of radians subtends an arc whose length s is s = rθ.
Hence, notice that linear speed is equal angular speed multiplied, not divided, by r.
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What is the displacement (pls answer )(right answers only plssss)
Answers
Answer:
I think its B
Step-by-step explanation:
north and south cancel out, he went in the shape of an n
The circumference of a sphere was measured to be 76 cm with a possible error of 0.5 cm. (a) Use differentials to estimate the maximum error in the calculated surface area. (Round your answer to the nearest integer.) cm2 What is the relative error
Answers
The maximum error in the calculated surface area is 24.19cm² and the relative error is 0.0132.
Given that the circumference of a sphere is 76cm and error is 0.5cm.
The formula of the surface area of a sphere is A=4πr².
Differentiate both sides with respect to r and get
dA÷dr=2×4πr
dA÷dr=8πr
dA=8πr×dr
The circumference of a sphere is C=2πr.
From above the find the value of r is
r=C÷(2π)
By using the error in circumference relation to error in radius by:
Differentiate both sides with respect to r as
dr÷dr=dC÷(2πdr)
1=dC÷(2πdr)
dr=dC÷(2π)
The maximum error in surface area is simplified as:
Substitute the value of dr in dA as
dA=8πr×(dC÷(2π))
Cancel π from both numerator and denominator and simplify it
dA=4rdC
Substitute the value of r=C÷(2π) in above and get
dA=4dC×(C÷2π)
dA=(2CdC)÷π
Here, C=76cm and dC=0.5cm.
Substitute this in above as
dA=(2×76×0.5)÷π
dA=76÷π
dA=24.19cm².
Find relative error as the relative error is between the value of the Area and the maximum error, therefore:
\(\begin{aligned}\frac{dA}{A}&=\frac{8\pi rdr}{4\pi r^2}\\ \frac{dA}{A}&=\frac{2dr}{r}\end\)
As above its found that r=C÷(2π) and r=dC÷(2π).
Substitute this in the above
\(\begin{aligned}\frac{dA}{A}&=\frac{\frac{2dC}{2\pi}}{\frac{C}{2\pi}}\\ &=\frac{2dC}{C}\\ &=\frac{2\times 0.5}{76}\\ &=0.0132\end\)
Hence, the maximum error in the calculated surface area with the circumference of a sphere was measured to be 76 cm with a possible error of 0.5 cm is 24.19cm² and the relative error is 0.0132.
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Two cards are randomly drawn.
What is the probability of drawing two blue cards if
the first card is not included in the second draw?
Simplify your answer completely.
5
Enter the number that
[?] belongs in the green box.
Hint: Multiply the probability of the 1st Event by the
probability of the 2nd Event to get your answer,
Enter
Answers
Answer: 5/14 ^^
Step-by-step explanation:
Fill in the Blank: a. The entire collection of objects being studied is called the ________________. b. A small subset from the set of all 2013 minivans is called a ________________. c. Consider the amount of sugar in breakfast cereals. This characteristic of breakfast cereal (objects) is called a ________________.
Answers
a. The entire collection of objects being studied is called the population.
b. A small subset from the set of all 2013 minivans is called a sample.
c. Consider the amount of sugar in breakfast cereals. This characteristic of breakfast cereal (objects) is called a variable.
a. Population: The population refers to the entire group or collection of objects, individuals, or units that are of interest in a study. It represents the complete set of items from which a sample is drawn. For example, if you are conducting a study on the heights of all adults in a particular country, the population would consist of every adult in that country.
b. Sample: A sample is a smaller subset or representative portion of the population. It is selected from the larger population with the intention of making inferences or generalizations about the population. Sampling is often done when studying an entire population is not feasible or practical. In the context of the example given, a sample of 2013 minivans could be randomly selected from the entire set of minivans produced in 2013.
c. Variable: A variable is a characteristic or attribute that can vary or take different values within a population or sample. In the given example of breakfast cereals, the amount of sugar is a variable. Variables can be quantitative, such as numerical measurements like weight or height, or qualitative, such as categories or labels like color or brand. In statistical analysis, variables are used to describe and analyze data, and they can be classified as independent variables (predictors) or dependent variables (outcomes).
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18 24 11 What is the area of the triangle shown? A.99 square units
В. 132 square units
C. 198 square units
D.264 square units
Answers
Answer: I think B
Step-by-step explanation:
Who could help me with this please and thank you
Answers
Answer:
the solutions are (1,-3) and (4,-6)
Alyssa sold half of her comic books and then bought 8 more. She now has
12. How many did she begin with ?
Answers
Answer:
8 Books
Step-by-step explanation:
First find how many she had before buying 8 more-
12 - 8 = 4
Then multiply that by 2
4 x 2 = 8
The had 8 books to begin with
plz help asap 14 points
Latoya created a factor tree and wrote the prime factorization of 90 shown below.
A factor tree of 90. 90 branches to 9 and 10. 9 branches to 3 and 3. 10 branches to 2 and 5. The equation is 90 = 2 times 3 times 5.
What is Latoya’s error?
She should not have found the factors of 9.
She should have included an exponent of 2 on the 3.
She should have included 9 and 10 in the prime factorization.
She should have started the tree with 2 times 45.
Answers
Answer:
B
Step-by-step explanation:
Latoya created a factor tree and wrote the prime factorization of 90 shown below.
A factor tree of 90. 90 branches to 9 and 10. 9 branches to 3 and 3. 10 branches to 2 and 5. The equation is 90 = 2 times 3 times 5.
What is Latoya’s error?
She should not have found the factors of 9.
She should have included an exponent of 2 on the 3.
She should have included 9 and 10 in the prime factorization.
She should have started the tree with 2 times 45.
Hope this helps!!!!!!
Answer:
B
Step-by-step explanation: make it brainleist plz