Solving Angular Momentum Homework: <L^2>,<(L^2)^2>, etc.

In summary, the question asks for the expectation values of <L^2>, <(L^2)^2>, <L_{z}>, <L_{z}^2>, \Delta L^2, and \Delta L_{z}. The equations for these values are provided and the attempt at a solution is shown. The values given in the attempt are correct, but there was a simple math mistake made in the calculation of the first two answers.
  • #1
stunner5000pt
1,461
2

Homework Statement


For the non stationary state

[tex] \Psi = \frac{1}{\sqrt{2}} \left(Psi_{100}+\Psi_{110}\right) = \frac{1}{\sqrt{2}} \left(R_{10}Y_{00}e^{-iE_{10}t/\hbar}+R_{11} Y_{10}e^{-iE_{11}t/\hbar}[/tex]

find [itex] <L^2>,<(L^2)^2>,<L_{z}>,<L_{z}^2>,\Delta L^2, \Delta L_{z}[/itex]

Homework Equations


[tex] <L^2>=\hbar^2 l(l+1)[/tex]
[tex] <(L^2)^2>=(\hbar^2 l(l+1))^2[/tex]
[tex] <L_{z}>=\hbar m_{l}[/tex]
[tex] <L_{z}^2>=(\hbar m_{l})^2[/tex]
[tex] \Delta x = \sqrt{<x^2>-<x>^2} [/tex]

The Attempt at a Solution


[tex]<L^2>=\frac{\hbar^2}{2} \left(0(0+1) + 1(1+1)\right)= \frac{3}{2} \hbar^2 [/tex]
the answer is supposed to be hbar^2 ... what am i doing wrong...

[tex] <(L^2)^2> = \frac{\hbar^4}{4} \left((0(0+1))^2+(1(1+1))^2\right) =\frac{5}{4} \hbar^2[/tex]

[tex] <L_{z}> = \frac{\hbar}{2} (0+0) = 0 [/tex]

[tex] <L_{z}^2} = \frac{hbar^2}{4} (0) = 0 [/tex]

are the values right... i fear that i am sorely mistaken about how to calculate the expectation values

any help would be greatly appreciated!
 
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  • #2
For your first answer, try 1*2=2, and 0*1=0.
Also 0*1=0 for the second answer.
Back to arithmetic 101.
 
Last edited:
  • #3
Meir Achuz said:
For your first answer, try 1*2=2, and 0*1=0.
Also 0*1=0 for the second answer.
Back to arithmetic 101.

OOPS

stupid me
 

What is angular momentum?

Angular momentum is a property of a rotating object and is defined as the product of its moment of inertia and angular velocity. It is a vector quantity that describes the rotational motion of an object.

How is angular momentum calculated?

Angular momentum (L) is calculated as the cross product of the position vector (r) and the linear momentum (p): L = r x p. It can also be calculated using the moment of inertia (I) and the angular velocity (ω): L = Iω.

What is the significance of and <(L^2)^2> in angular momentum homework?

and <(L^2)^2> are operators used to represent the square of the angular momentum in quantum mechanics. They are important in solving problems related to the quantization of angular momentum and determining the allowed values of the angular momentum of a particle.

What are the units of angular momentum?

The SI unit of angular momentum is kilogram meter squared per second (kg·m^2/s). In quantum mechanics, the units of angular momentum are expressed as h-bar (ħ), which is equal to 1.054571817 × 10^-34 joule seconds (J·s).

How is angular momentum conserved in a system?

According to the law of conservation of angular momentum, the total angular momentum of a system remains constant if there are no external torques acting on the system. This means that the angular momentum of a system cannot be created or destroyed, but can only be transferred between different parts of the system.

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