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Fundamentals Of Physics
Found in: Page 1247

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Short Answer

If orbital angular momentum is measured along, say, a z-axis to obtain a value for Lz, show that role="math" localid="1661497092782" (Lx2+Ly2)1/2=[I(I+1)-mI2]1/2ħ is the most that can be said about the other two components of the orbital angular momentum.

It is proved for the other two components of the orbital angular momentum is


See the step by step solution

Step by Step Solution

Step 1: The given data:

The orbital angular momentum L is measured along a z-axis to get a value Lz.

Step 2: Understanding the concept of angular momentum:

In quantum mechanics, angular momentum is a vector operator with well-defined commutation relations between its three components.

Using the concept of orbital angular momentum and the value of the momentum in the z-axis in the resultant value of the momentum, define the required expression of the x- and y-component of the angular momentum.


The magnitude of the orbital angular momentum in terms ħ of is,

L=II+1ħ ….. (1)

The z-component of the orbital angular momentum is,

Lz=mIħ ….. (2)

The resultant value of the angular momentum is,

L2=Lx2+Ly2+Lz2 ….. (3)

Step 3: Calculation to get the required equation:

Substituting the values of equations (1) and (2) in equation (3), the required equation to be proved as follows:

I(I+1)ħ2=Lx2+Ly2+mI ħ2Lx2+Ly2=I(I+1)ħ2-mI ħ2Lx2+Ly2=I(I+1)-mI21/2ħ

For a given value of I, the greatest value of m1 is , so the smallest value of Lx2+Ly2 is ħI.

Now, the smallest possible value of m1 is zero, thus the largest possible value of is .

Lx2+Ly2 is ħII+1

So, the range is given by:

ħILx2+Ly2 II+1

Hence, the given equation is proved.

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