Solve L=2 Atom Problem: Find Min (L_x)^2 + (L_y)^2

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In summary, the smallest value of (L_x)^2 + (L_y)^2 for an electron in an l=2 state is 2h(bar)^2. This is determined by the conservation of total angular momentum, where L_z has a maximum value of 4h_bar^2 and is related to L_x and L_y states.
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PsychonautQQ
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Homework Statement


For an electron in an atom that is an l=2 state, find the smallest value of (L_x)^2+(L_y)^2.


Homework Equations





The Attempt at a Solution


okay so L = (l(l+2))^(1/2)

and that squared with l =2 would be 6h(bar)^2

kind of running into a wall here..
Or is it the maximum value of (L_x)^2 + (L_y)^2 occurs when (L_z)^2 is a maximum value? and L_z = m_l*h(bar).. which would have a maximum value of 4h_bar^2 so therefore L_x^2 + L_y^2 would be 2h(bar)^2?

Ps I need to learn LaTex X_x
 
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l=2 tells you the Lz state.
How do the Lx and Ly states relate to that?
Hint: conservation of total angular momentum.
 

FAQ: Solve L=2 Atom Problem: Find Min (L_x)^2 + (L_y)^2

What does "L=2 Atom Problem" refer to?

The L=2 Atom Problem is a quantum mechanical problem that involves finding the minimum value of the squared angular momentum (L) of an atom with a specified total angular momentum value of 2. It is commonly used in atomic physics to understand the energy levels and properties of atoms.

What is the significance of finding the minimum value of (L_x)^2 + (L_y)^2?

In quantum mechanics, the squared angular momentum components (L_x)^2 and (L_y)^2 represent the projections of the total angular momentum in the x and y directions, respectively. Finding the minimum value of their sum is important because it indicates the lowest possible energy state of the atom and provides valuable information about its structure and behavior.

How is the L=2 Atom Problem solved?

The L=2 Atom Problem can be solved using mathematical techniques such as matrix diagonalization or the Wigner-Eckart theorem. These methods involve manipulating the wave function of the atom and using symmetry properties to determine the minimum value of (L_x)^2 + (L_y)^2.

What are the practical applications of solving the L=2 Atom Problem?

Solving the L=2 Atom Problem has a wide range of applications in fields such as atomic and molecular physics, quantum chemistry, and materials science. It can help in understanding the properties of atoms and molecules, predicting their behavior, and developing new technologies based on quantum mechanics.

Are there any limitations to the L=2 Atom Problem?

Like any other mathematical model, the L=2 Atom Problem has its limitations. It assumes that the atom is in a perfect vacuum and does not take into account external factors such as magnetic fields or other atoms in its vicinity. Additionally, it is a simplified version of the more complex L=2 Atom Problem with spin, which takes into account the intrinsic spin of electrons in the atom.

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