The spherical harmonics Y0,0, Y2,-1, and Y3,+3 satisfy the Schrödinger equation for a particle free to rotate in three dimensions, and have energies and angular momenta of E=0 and Lz=0, E=6.
(a) For Y0,0, the wave function ψ is proportional to Y0,0 and is independent of θ and φ. Therefore, the Laplacian operator acting on ψ reduces to:
∇^2ψ = (1/r^2) ∂/∂r (r^2 ∂/∂r) Y0,0 = -l(l+1) Y0,0
where l = 0 is the angular momentum quantum number. Substituting this into the Schrödinger equation gives:
(-ħ^2/2μ) (-l(l+1)) Y0,0 = E Y0,0
which has the solution E = 0 and angular momentum Lz = 0.
(b) For Y2,-1, the wave function ψ is proportional to Y2,-1 and depends on θ and φ. Therefore, the Laplacian operator acting on ψ reduces to:
∇^2ψ = (1/r^2) ∂/∂r (r^2 ∂/∂r) Y2,-1 - (2/r^2 sinθ) ∂/∂φ Y2,-1 = -l(l+1) Y2,-1
where l = 2 is the angular momentum quantum number. Substituting this into the Schrödinger equation gives:(-ħ^2/2μ) (-6) Y2,-1 = E Y2,-1which has the solution E = 6(ħ^2/2μ) and angular momentum Lz = -ħ.
(c) For Y3,+3, the wave function ψ is proportional to Y3,+3 and depends on θ and φ. Therefore, the Laplacian operator acting on ψ reduces to:
∇^2ψ = (1/r^2) ∂/∂r (r^2 ∂/∂r) Y3,+3 + (6/r^2 sinθ) ∂/∂φ Y3,+3 = -l(l+1) Y3,+3
where l = 3 is the angular momentum quantum number. Substituting this into the Schrödinger equation gives:
(-ħ^2/2μ) (-12) Y3,+3 = E Y3,+3which has the solution E = 12(ħ^2/2μ) and angular momentum Lz = +3ħ.
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To confirm that the spherical harmonics Y0,0, Y2,-1, and Y3,+3 satisfy the Schrödinger equation for a particle free to rotate in three dimensions, we need to substitute them into the equation and see if they hold true. Once we do that, we can solve for the energy and angular momentum in each case.
The Schrödinger equation involves the dimensions of position, momentum, and time, and it describes the behavior of quantum particles. For particles free to rotate in three dimensions, the equation involves angular momentum and its associated operators. The solutions for the spherical harmonics satisfy the Schrödinger equation and have well-defined energy and angular momentum values. By calculating these values for Y0,0, Y2,-1, and Y3,+3, we can better understand the behavior of quantum particles in three-dimensional space.
To confirm that the spherical harmonics Y0,0, Y2,-1, and Y3,+3 satisfy the Schrödinger equation for a particle free to rotate in three dimensions, we must first examine the equation, which describes the relationship between the energy (E) and the angular momentum (L) of the system.
For a particle free to rotate in 3D, the Schrödinger equation takes the form: Hψ = Eψ, where H is the Hamiltonian operator, ψ represents the wavefunction, and E is the energy. Spherical harmonics are solutions to this equation when the Hamiltonian only involves the angular momentum operator.
(a) Y0,0: With L=0 and M=0, the energy and angular momentum are E=0 and L=0.
(b) Y2,-1: With L=2 and M=-1, the energy is E=2(2+1)ħ²/2I, and the angular momentum is L=ħ√(2(2+1)).
(c) Y3,+3: With L=3 and M=3, the energy is E=3(3+1)ħ²/2I, and the angular momentum is L=ħ√(3(3+1)).
In all three cases, the spherical harmonics satisfy the Schrödinger equation, with the energy and angular momentum being proportional to their respective quantum numbers.
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[tex] \sqrt{ x} = \sqrt{2} + 1 \: then \: x = [/tex]
helpppp asap
edit:help guys i have examm
Answer:
3+2√2Step-by-step explanation:
Given the expression
√x = √2 + 1
To get x, we will have to square both sides
(√x)² = (√2 + 1)²
x = (√2 + 1)(√2 + 1)
x = √4+√2+√2+1
x = 2+2√2+1
x = 3+2√2
Hence the value of x is 3+2√2
Which expression can be used to find the volume of the sphere? A sphere with radius 9 inches. V = four-thirds pi r squared = four-thirds (3.14) (9) squared V = four-thirds pi r cubed = four-thirds (3.14) (9) cubed V = four-thirds pi r cubed = four-thirds (3.14) (18) cubed V = four-thirds pi r cubed = four-thirds (3.14) (18) cubed Mark this and retur
Answer:
V = four-thirds pi r cubed = four-thirds (3.14) (9) cubed
Step-by-step explanation:
You have 49 dimes and quarters. You have 7 more quarters than dimes.How much money you have in quarters and dimes
Answer:
Step-by-step explanation:
Because you have seven more quarters than dimes, you have 28 quarters, and 21 dimes, which equals 49 dimes and quarters.
You have $9.10
I hope this helps!
Thanks!
A home meal delivery business delivers 34.5 pounds of food each day. How many ounces of food will they deliver in 4 days? (1 lb.= 16 oz) fill in the blank they will deliver ______ ounces of food in 4 days.
Answer:
They will deliver 2208 ounces of food in 4 days.
Step-by-step explanation:
Multiply 34.5 by 16 to get the ounces and got 552. Then multiply 552 by 4 and you'll get the answer.
:))
Please help with this it is due today
Answer:
24
Step-by-step explanation:
Someone please help me!! I will mark as brainliest
Answer:
if it's not on a test, just guess
Step-by-step explanation:
Which statement is true for brainliest
Answer:
<A≅<Z
Step-by-step explanation:
Hope This Helps!
if I have 100 baskets and I have 400 hundred apples how many apples will go into each basket
Answer: 4 apples will go into each basket if you want all the baskets to be filled with an equal amount.
Step-by-step explanation:
100x=400
dividing through by 100 we get
x=4
can someone help me how to calculate 15% of R200
Answer:
30
Step-by-step explanation:
formula :15÷100×200
8th grade math. Solve and check. 2x + 3x + 5 = 4(x - 2)
Answer:
x=−13
Step-by-step explanation:
Combine the like terms:
2+3+5=4(−2)5+5=4(−2)
Distribute:
5+5=4(−2)
5+5=4−8
Subtract 5 from both sides
5+5=4-85+5−5=4−8−5
Then simplify:
x=−13
Answer:
x=−13
Solve -3.5=x-4 for x using the multiplication propertiy of equality
Answer:
Hello! :) have a good day!
X= 0.5
Answer:
x = 0.5
Step-by-step explanation:
how can you tell the difference between circumcenter and incenter?
Beniceo Skis 748yd in 3 mins. At this rate, How many miles does beniceo ski in an hour? (1 mi= 1,760 yd). mi=miles yd= yards
Answer:
8.5 miles
Step-by-step explanation:
3x20=60, 60 minutes also being an hour so if you multiply 748 by 20 then it gives you 14,960 annndd now you have to divide by the amount of yards in a mile 14,960/1,760=8.5
A jungle has 578 monkeys.
It also has 153 birds.
How many more monkeys than birds are there?
Monkeys
X Х $ ?
Answer:
425 monkeys
Step-by-step explanation:
Subtract the number of birds from the number of monkeys to find the difference between the two.
Jack invested $7,600 in an account paying an interest rate of 6\tfrac{1}{4}6 4 1 % compounded continuously. John invested $7,600 in an account paying an interest rate of 6\tfrac{3}{4}6 4 3 % compounded quarterly. After 19 years, how much more money would John have in his account than Jack, to the nearest dollar?
Answer: John would have $1371.33 more in his account than Jack.
Step-by-step explanation:
Formula to calculate amount after t years if compounded continuously:
[tex]A=Pe^{rt}[/tex]
Formula to calculate amount after t years if compounded quarterly:
[tex]A=P(1+\dfrac{r}{4})^{4t}[/tex]
, where P= Principal amount, r= rate of interest.
Given: P= $7,600
t=19 years
For Jack, [tex]r=6\dfrac{1}{4}\%=\dfrac{25}{4}\%=\dfrac{25}{400}=0.0625[/tex]
For John, [tex]r=6\dfrac{3}{4}\%=\dfrac{27}{4}\%=\dfrac{27}{400}=0.0675[/tex]
Amount in Jack's account after 19 years= [tex]7600e^{0.0625(19)}[/tex]
[tex]=7600e^{1.1875}\\\\=7600(3.27887376794)\\\\\approx\$24,919.44[/tex]
Amount in John's account after 19 years= [tex]7600e^{0.0675(19)}[/tex]
[tex]=7600(1+0.0675)^{19}\\\\=7600(3.45931245291)\\\\\approx\$26,290.77[/tex]
Difference = 26290.77-24919.44 = $ 1,371.33
John would have $1371.33 more in his account than Jack.
Answer:668
Step-by-step explanation: I just did it.
GIVING BRAINLIEST IM TIMEDDDD
Answer:its the 2nd one
Step-by-step explanation: i remeber because i took this quiz
Help am i correct please Ehejahrs
Simplify the following
(1 1/2) ^-2
Answer:
1/2.25 or 4/9
Step-by-step explanation:
What value of x makes the equation below true 5(x − 8) + 26 = −9? A 1 B -1 C 15 D -15
Which equation correctly uses the law of cosines to solve for the length s?
Answer: Choice D
s^2 = 9^2 + 10^2 - 2(9)(10)cos(100)
==========================================================
Explanation:
Consider the pythaorean theorem
a^2 + b^2 = c^2
which is the same as saying
c^2 = a^2 + b^2
If we tack on "-2ab*cos(C)", then we end up with
c^2 = a^2 + b^2 - 2ab*cos(C)
which is one form of the law of cosines. The side c is opposite angle C, and the two are on opposite sides of the equation. The 'a's and 'b's are on the right side with the angle C.
Note if C = 90 degrees, then cos(C) = cos(90) = 0 which means the whole "-2ab*cos(C)" term goes to 0 and we'd be left with the pythagorean theorem again.
In this case, we replace c with s, replace 'a' with 9, replace b with 10, and C with 100.
Answer:
Option (4)
Step-by-step explanation:
Cosine rule states,
(UT)² = (SU)² + (ST)² - 2(SU)(ST)cos(∠UST)
Here, UT is the opposite side of angle UST and two sides SU and ST are the adjacent sides.
∠UST is the included angle between SU and ST.
By this formula,
s² = 9² + 10² - 2(9)(10)cos(100°)
Therefore, Option (4) will be the answer.
7 The table below shows the population of
different regions in 1985 and in 2005.
Place
Entire world
China
Population in Population
1985
in 2005
4.9 X 109 6.4 X 109
1.1 X 109 1.3 X 109
7.6 x 108 1.1 x 10
2.4 x 108 3.0 X 108
8.9 X 106 7.2 X 106
India
United States
Bulgaria
a Determine the ratio of the population
of the entire world to that of China in
2005, giving your answer to the nearest
integer
Answer:
In this table the graphic shows that...
Step-by-step explanation:
...the population increased 5,9 x 10 6.
The ratio is 64:13
What is ratio?Ratio is the comparison of two quantities which is obtained by dividing the first quantity by the other. If a and b are two quantities of the same kind and with the same units, such that b is not equal to 0, then the quotient a/b is called the ratio between a and b. Ratios are expressed using the symbol of the colon (:). This means that ratio a/b has no unit and it can be written as a : b
Given data:
Population of china in 2005 =1.3*10^9
population of entire world in 2005 = 6.4*10^9
Now,
ratio= 6.4*10^9/ 1.3*10^9
ratio= 64/13 or 64:13
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What is the slope of the linear equation represented by the table. NEEDED PLSS
Answer:28/4
Step-by-step explanation:
Change of y over the change of x.
Y is +28
X is +4
Pete is choosing his clothes for the day. His bottoms will be either cargo shorts(C) or jeans (J). His top will be T shirt (T), sweatshirt (S), or long sleeved shirt (L)
The question is incomplete. Here is the complete question.
Pete is choosing his clothes for the day. His bottoms will be either cago shorts (C) or jeans (J). His top will be T-shirt (T), sweatshirt (S) or long sleeved shirt (L).
Which is a complete list of the possilbe outfits he can choose from?
CT, CS, CL, JT, JS, JL
CJ, CT, CS, CL
CJT, CJS, CJL
CT, TC, JS, SJ, CL, LC
Answer: CT, CS, CL, JT, JS, JL
Step-by-step explanation: The Fundamental Counting Principle is a method of determining a number of ways independents events can occur, i.e., it states that if there are t ways of doing something and w ways of doing another, then there are t x w ways of doing both things.
For example:
Peter has 2 choices of bottom and 3 choices of top. So, according to the principle, he will have 2 x 3 choices of clothing.
And the choices are:
shorts and t-shirt = CT
shorts and sweatshirt = CS
shorts and long sleeved = CL
jeans and T-short = JT
jeans and sweatshirt = JS
jeans and long sleeved = JL
The order he chooses doesn't matter.
So, the right alternative is CT, CS, CL, JT, JS, JL.
Solve the pair of linear equations....
Please help me by solving this!
Hangi sayının 10 eksiğinin 7 katı 70'tir ?
Simplify: i^69
Please help fast
Answer: it's just "i"
Step-by-step explanation:
25 POINTS!!!!! HELP NEEDED
(05.01 MC)
The graph shows the prices of different numbers of bushels of corn at a store in the
current year. The table shows the prices of different numbers of bushels of corn at
the same store in the previous year.
Current Year
98
70
Price of
Com
56
(dollars) 42
28
14
0 2 4 6 8 10 12 14
Number of Bushels
Previous Year
Number of Bushels Price of Corn (dollars)
2
10
4.
20
6
30
8
40
PART A:
To find the rate of change, put change in y over change in x. In order to first find the changes, you must subtract two y values from each other and two x values from each other, like this:
Change in y = 20-10
Change in x = 4-2
Then you can put these over each other:
20-10/4-2
Then simplify
20-10/4-2 = 10/2 = 5/1 = 5
So, this means the rate of change is 5. $5 per bushel.
If you also need part B it would be helpful to see the other chart because it's hard to understand in the writing you included. :) Comment and let me know if you post it so I can help you out!
Part A: 7 dollars
- The graph shows that 2 bushels is 14 dollars, because the graph has a straight line, the rate remains constant you must divide the sides by two to find the amount for one bushel which would be 7 dollars.
- And to double check you would divide 28 by 4, 42 by 6, 56 by 8, 70 by 10, 84 by 11, and 98 by 12. they all equals 7 dollars So, the equation would be y = 7x.
- 7 can be known as the slope and rate of change.
Part B:2 dollars
- The years before show 2 bushes was 10 dollars
-so you would divide 10 by two you would get five To double check you would do the same thing with 20 by 4, 30 by 6 and 40 by 8, and they all equal 5 dollars.
-if you use this answer be sure to reword it so you dont get copy write and integrity. have a great day or night.
Please Help!
if these two shapes are similar, what is the measure of missing length q?
Answer:
40
Step-by-step explanation:
96 ÷ 60 = 1.6
25 × 1.6 = 40
Disclaimer: Yay, okay. The person above me got the same answer so this is right.
BRAINLIEST GOES TO THE CORRECT ANSWER!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answer:
11.99
Step-by-step explanation:
Ethan is designing a new board game, and is trying to figure out all the possible
outcomes. How many different possible outcomes are there if he spins a spinner with
three equal-sized sections labeled Walk, Run, Stop and rolls a fair die in the shape of
a cube that has six sides labeled 1 to 6?
Answer:
18
Step-by-step explanation:
3x6