Note : The answer should be typed.

6.2. Spin-wave exercise (a) The Holstein-Primakoff transformation defines spin operators via creation and annihilation operators of a harmonic oscillator, a and a' with the standard commutation relation [a , a'] = 1: $1 = V25 a , 9 = 0 V/25 28 Show that the standard commutation relation [St, 9-] - 25: holds. (b) In the lecture, we showed that the operators a and at describe magnon excitations that follow Buso- Einstein statistics with the occupation numbers 1 Calculate magnetization as a function of temperature, M(T) - MO) 1 *5 )=) and show that at low temperatures (Aux o k") it follower Bloch's 7: law. Hint: transform the summation inco an integral and change the integration variable such that temperature goes Into the integration variable. Then the integral becomes T-independent, and all temperature dependence remains in the pro-factor