A hydrogen atom is in a state with quantum number n=3 . In...

A hydrogen atom is in a state with quantum number $n=3 .$ In the quantum-mechanical description of the atom, what is the largest possible magnitude of the orbital angular momentum? What is the magnitude of the orbital angular momentum of the electron in the Bohr-model description of the atom? What is the percentage difference between these two values? (b) Answer the same questions as in part (a) for $n=30$.

Key Concepts

Quantum Numbers

Quantum numbers are sets of numerical values that provide solutions to the Schrödinger equation and characterize the properties of an electron in an atom. The principal quantum number n primarily determines the energy level and size of the orbital, while the orbital quantum number l, which can range from 0 to n-1, determines the orbital angular momentum and the shape of the electron cloud. These numbers are fundamental in both the quantum mechanical model and the Bohr model, though they play different roles regarding angular momentum quantization.

Orbital Angular Momentum in Quantum Mechanics

In quantum mechanics, the orbital angular momentum of an electron is quantized and given by the formula L = ??[l(l+1)], where l is the orbital quantum number and ? is the reduced Planck constant. The maximum possible angular momentum for a given energy level is obtained when l is at its maximum value of n - 1, leading to L_max = ??[(n-1)(n)]. This quantization reflects the intrinsic uncertainties and wave nature of the electron and underscores the departure from classical intuitions.

Orbital Angular Momentum in Bohr Model

The Bohr model, an early quantization scheme, assumes that the electron’s orbital angular momentum is quantized in integral multiples of the reduced Planck constant, expressed as L = n?, where n is the principal quantum number. This simpler rule, though historically significant and successful for explaining certain spectral lines, does not capture the more accurate quantum mechanical behavior which uses the additional factor of ?[l(l+1)] when l is maximized.

Percentage Difference Calculation

The percentage difference is a way to quantify the relative discrepancy between two values by taking the absolute difference, dividing it by one reference value (usually the quantum-mechanical result in this context), and then multiplying by 100. This calculation provides a clear measure of how much the Bohr model's prediction deviates from the more precise quantum mechanical prediction for the orbital angular momentum.

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