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Found in: Page 278

### Modern Physics

Book edition 2nd Edition
Author(s) Randy Harris
Pages 633 pages
ISBN 9780805303087

# Question: Section 7.5 argues that knowing all three components of would violate the uncertainty principle. Knowing its magnitude and one component does not. What about knowing its magnitude and two components? Would be left any freedom at all and if so, do you think it would be enough to satisfy the uncertainly principle?

Only the magnitude of L and the Z-component of L can be found accurately.

See the step by step solution

## Step 1: Definition of the Uncertainty principle

The Uncertainty principle states that the momentum and position of a particular particle cannot be measured with great accuracy. In general, it explains that uncertainty will be there in measuring a particular variable for a specific particle.

## Step 2: Determination of the violation of uncertainty principal and magnitude if two components are known

Write the expression for the sum of the square of the components.

${L}_{x}^{2}+{L}_{y}^{2}+{L}_{z}^{2}={\left|\text{L}\right|}^{2}$

Here, ${\left|\text{L}\right|}^{}$ is the magnitude of angular momentum vector, LX is the x-component of angular momentum vector, LY is the y-component of angular momentum vector, and LZ is the z-component of angular momentum vector.

It is known that LZ can be known with certainty, by the equation:${L}_{Z}={m}_{l}\hslash$ , and is also quantized and can be calculated by the following equation.

$L=\sqrt{l\left(l+1\right)}\hslash$

Here, is the Plank’s Constant, and is the azimuthal quantum number

So, does not commute with LX or LY . Hence, if relation is${L}_{x}^{2}+{L}_{y}^{2}+{L}_{z}^{2}={\left|\text{L}\right|}^{2}$ being used to calculate the value of LX and , LY then it will be uncertain.

Thus, only the magnitude of L and the z-component of L can be found accurately.

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