Harmonic Oscillator in Field

A particle of mass m and charge e moves in one dimension. It is attached to a spring of constant k and is initially in the ground state (n = 0) of the harmonic oscillator. An electric field E is switched on during the interval 0 < t < τ, where τ is a constant.

a) What is the probability of the particle ending up in the n = 1 state?

b) What is the probability of the particle ending up in the n = 2 state?

SOLUTION

Now we label the eigenstates with the index n for the harmonic oscillator state of energy E_n = hω(n + 1/2) and write the equation satisfied by the time-dependent amplitudes a_n(t),

(1)

(2)

We need to evaluate the matrix element ⟨n|x|m⟩ of x between the states |n⟩ and |m⟩ of the harmonic oscillator. It is only nonzero if m = n ±1. In terms of raising and lowering operators,

(3)

(4)

(5)

(6)

a) If the initial state is n = 0 at t = 0, then the amplitude of the n = 1 state for (t < τ) is given by

(7)

(8)

The last equation is the probability of ending in the state n = 1 if the initial state is n = 0. This expression is valid as long as it is less than 1 or if 2e|E|X₀ < hω.

b) The n = 2 state cannot be reached by a single transition from n = 0 since the matrix element ⟨2|x|0⟩ = 0. However, n = 2 can be reached by a two-step process. It can be reached from n = 1, and n = 1 is excited from n = 0. The matrix element is so we have that

(9)

(10)

Note that P₂ = P₁²/2!. Similarly, one can show that P_n = P₁ⁿ/n! However, the total probability, when summed over all transitions, cannot exceed 1. Therefore, we define a normalized probability

(11)

مواضيع ذات صلة

مصطلحات الفيزياء النووية

إزاحة حمراء REDSHIFT

المركبات النانوية (Nanocomposites)

مولد فان دي جراف الالكتروستاتي

التطويرات التقنية الحديثة في المعجلات الالكتروستاتية

معجلات الالكترونات الالكتروستاتية

معجلات الفان دي جراف الترادفية

الحد من زيادة الطاقة في معجل فان دي جراف

أنبوبة التعجيل قي معجلات الجسيمات من نوع فان دي جراف

ارتكاز معجل فان دي جراف

مبدأ تشغيل معجل الفان دي جراف

القضايا الأخلاقية المتعلقة بتكنولوجيا النانو