Question

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?

Short answer

Answer

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

Step-by-step solution

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.

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, L_X is the x-component of angular momentum vector, L_Y is the y-component of angular momentum vector, and L_Zis the z-component of angular momentum vector.

It is known that L_Z 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.

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

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

So, does not commute withL_X or L_Y . 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 L_X and , L_Y then it will be uncertain.

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