Show, by induction or otherwise, that there are n^2 stationary states of hydrogen with energy E=

step 1 In answering this question, we need to recall that the ground state energy of a helium atom E is

Show, by induction or otherwise, that there are n^2...

Key Concepts

Stationary States

Stationary states are quantum states with all observables independent of time since they are eigenstates of the Hamiltonian. In systems like the hydrogen atom, these states correspond to fixed energy levels, which is central to understanding the atom’s spectral lines and energy quantization.

Quantum Numbers

Quantum numbers are used to label the unique stationary states in quantum systems. For the hydrogen atom, the principal quantum number (n) determines the energy level, while other quantum numbers such as the orbital angular momentum quantum number (l) and the magnetic quantum number (m) specify the state’s angular properties. These numbers together uniquely define each state within a given energy level.

Degeneracy

Degeneracy in quantum mechanics refers to the phenomenon where different quantum states share the same energy. In the hydrogen atom, for a given principal quantum number n, multiple combinations of l and m yield the same energy eigenvalue. Counting these combinations shows that there are exactly n² distinct stationary states for each energy level with energy E = -R/n².

Summation Techniques and Induction

Counting the number of degenerate states often involves summing over the allowed values of the quantum numbers. For the hydrogen atom, summing over l from 0 to n-1, the number of m states associated with each l is (2l+1), and the total sum ?(2l+1) equals n². This summation can be verified by mathematical induction or direct calculation to establish the n² degeneracy of each energy level.

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