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Chapter 7: Q16CQ (page 279)

Can the transition 2s鈫�1s in the hydrogen atom occur by electric dipole radiation? The lifetime of the 2 s is known to be unusual. Is it unusually short or long?

Short Answer

Expert verified

Transition can鈥檛 occur by the electric dipole radiation. The electron will spend relatively long time in the 2 s before transitioning to the 1s. Thus, it is anticipated that the lifetime of the 2 s state be unusually long.

Step by step solution

01

Significance of electric dipole radiation

The oscillating electric dipole is possibly the vital source for the electromagnetic radiations. The electric dipole is placed in a particular electric field and given with some disturbance to produce oscillations, that is responsible for producing the electromagnetic radiations.

02

To determine transition of hydrogen atom from  occur by electric dipole radiation

From the selection rules of the electric dipole radiation, the change of / in the transition must be 1 or -1. Thus, the 2s鈫�1stransition can't occur by the electric dipole radiation, since the change in / would be zero.

Since that the most efficient radiation is given by an oscillating electric dipole and it is known that the 2s鈫�1stransition can't occur by electric dipole radiation. It is expected that the electron will spend relatively long time in the 2 s before transitioning to the 1s via other inefficient (unlike) types of radiation. Thus, it is anticipated that the lifetime of the 2s state be unusually long.

Therefore, transition can鈥檛 occur by the electric dipole radiation. The electron will spend relatively long time in the 2s before transitioning to the 1s. Thus, we anticipate that the lifetime of the 2s state be unusually long.

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Most popular questions from this chapter

A particular vibrating diatomic molecule may be treated as a simple harmonic oscillator. Show that a transition from that n=2state directly to n=0ground state cannot occur by electric dipole radiation.

Question: A particle is trapped in a spherical infinite well. The potential energy is 0for r < a and infinite for r > a . Which, if any, quantization conditions would you expect it to share with hydrogen, and why?

For states where l = n - t the radial probability assumes the general form given in Exercise 54. The proportionality constant that normalizes this radial probability is given in Exercise 64.

(a) Show that the expectation value of the hydrogen atom potential energy is exactly twice the total energy. (It turns out that this holds no matter what l may be)

(b) Argue that the expectation value of the kinetic energy must be the negative of the total energy.

Classically, it was expected that an orbiting electron would emit radiation of the same frequency as its orbit frequency. We have often noted that classical behaviour is observed in the limit of large quantum numbers. Does it work in this case? (a) Show that the photon energy for the smallest possible energy jump at the 鈥渓ow-n-end鈥� of the hydrogen energies is 3|E0|/n3, while that for the smallest jump at the 鈥渉igh-n-end鈥� is 2|E0|/n3, where E0is hydrogen鈥檚 ground-state energy. (b) Use F=ma to show that the angular velocity of a classical point charge held in orbit about a fixed-point charge by the coulomb force is given by 蝇=e2/4蟺蔚0mr3. (c) Given that r=n2a0, is this angular frequency equal to the minimum jump photon frequency at either end of hydrogen鈥檚 allowed energies?

The Diatomic Molecule: Exercise 80 discusses the idea of reduced mass. Classically or quantum mechanically, we can digest the behavior of a two-particle system into motion of the center of mass and motion relative to the center of mass. Our interest here is the relative motion, which becomes a one-particle problem if we merely use 渭for the mass for that particle. Given this simplification, the quantum-mechanical results we have learned go a long way toward describing the diatomic molecule. To a good approximation, the force between the bound atoms is like an ideal spring whose potential energy is 12kx2, where x is the deviation of the atomic separation from its equilibrium value, which we designate with an a. Thus,x=r-a . Because the force is always along the line connecting the two atoms, it is a central force, so the angular parts of the Schr枚dinger equation are exactly as for hydrogen, (a) In the remaining radial equation (7- 30), insert the potential energy 12kx2and replace the electron massm with 渭. Then, with the definition.f(r)=rR(r), show that it can be rewritten as

-魔22渭d2dr2f(r)+魔2I(I+1)2渭r2f(r)+12kx2f(r)=Ef(r)

With the further definition show that this becomes

-魔22渭d2dx2g(x)+魔2I(I+1)2渭(x+a)g(x)+12kx2g(x)=Eg(x)

(b) Assume, as is quite often the case, that the deviation of the atoms from their equilibrium separation is very small compared to that separation鈥攖hat is,x<<a. Show that your result from part (a) can be rearranged into a rather familiar- form, from which it follows that
E=(n+12)魔k渭+魔2I(I+1)2渭a2n=0,1,2,...I=0,1,2,...

(c)

Identify what each of the two terms represents physically.
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