Final answer:
The smallest angle that the angular momentum vector can make with the z-axis for an electron with quantum number l=2 is found using the magnetic quantum numbers. For m₁=+2, the angle would be the smallest, approaching 0 degrees.
Step-by-step explanation:
The question pertains to the calculation of the smallest angle that the angular momentum vector ℓ can make with the z-axis given the quantum number l for an electron. To find this, we use the concept of quantized angular momentum in quantum mechanics.
For an electron with a quantum number l=2, it can have multiple magnetic quantum numbers m₁, ranging from -l to +l. Therefore, for l=2, m₁ can be -2, -1, 0, 1, or 2. Each of these values corresponds to a different angle with the z-axis.
The smallest angle corresponds to the maximum value of m₁, which is 2 in this case. The formula cos 0 = Lz/L uses the magnetic quantum number m₁ and the total angular momentum to calculate the cosine of the angle. For m₁=+1, the value of cos 0 would be 0.707, using the provided ratios and simplifying the formulas given. This corresponds to an angle whose cosine is 0.707, which through the inverse cosine function can be determined to be approximately 45 degrees. However, for m₁=+2, the angle would be even smaller, approaching 0 degrees as the z-component of angular momentum becomes maximally aligned with the z-axis.