Find the average energy E for (a) An n-state system, in which a given state can have energy 0,,2,...,n . (b) A harmonic oscillator, in which a given state can have energy 0,,2,...(i. e., with no upper limit).

a♦1  E_average = n E₀ / 2 , b) E_average= infinity

Explanation:

The energy values form an arithmetic series, whose sum is

          S = n (a₁ + aₙ) / 2 = n (2a₁ + (n-1) r)/ 2

Where n is the number of terms, a₁ is the first term, aₙ the last term and r is the difference between two consecutive numbers in the series

          r = 2E₀ - 0 = 2E₀

Therefore the sum is

       S = n (0 + n E₀) / 2

      S = n² E₀ / 2

The average value is

         E_average = S / n

         E_average = n E₀ / 2

b) the case of harmonic oscillation

We have two possibilities.

- if we take a finite number and terms gives the same previous value

- If we take an infinite number of fears the series gives infinity and the average is also infinite

          E_average= infinity