8 Application to H atom

N. Panchapakesan and J.D. Anand

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    Learning Outcomes

  • Learn how selection rules for transitions are obtained by evaluating matrix elements and finding when they are non zero.
  • Learn how the life time of an excited state of hydrogen atom is calculated.
  • Become familiar with energy values of the states of hydrogen atom.
  • Learn a bit more about calculating with natural units.

    2. Introduction

 

You have studied the energy spectrum of hydrogen atom. We would like to apply the framework developed so far to the concrete case of hydrogen atom. We shall consider first when a transition from the one state to another state is allowed. The states are labeled by total quantum number n, the orbital quantum number and the azimuthal or magnetic quantum number m, If the initial state has values of n, and m and final state has values of n’,   ′ and m’ , we find that change =± 1 and m =0. ± 1 are only allowed. These are called selection rules and are derived by using the properties of spherical harmonics or associated Legendre functions.

 

We next study the specific transition in hydrogen from the state 2P ( =1, the first excited state ) to the ground state 1S ( =0 ). As = -1 this transition is allowed, so also for change in m. We carry out the calculations using the well known expressions for the wave functions for both the states and obtain the transition probability. The lifetime of the excited state is just the inverse of this value.

 

The calculations are specially simple if we use the natural units in which ħ= c=1. We discuss the use of such units in calculations in the last section more fully.

 

??????????? ?? ???????? ????

 

3.1 Selection Rules

 

In this section we apply our formalism to the electron bound in the hydrogen atom. We consider the structure of the matrix element

 

⟨?|?|?⟩ − − − − − − − − − − − − − − − − − (3.1)

 

Where the initial state A is characterized by the wave function

 

⟨?|?⟩ = ???(?)???(?, ?) − − − − − − − − − − − − − (3.2)

 

The final sate is characterized by

 

⟨?|?⟩ = ??′?′ (?)??′?′ (?, ?)− − − − − − − − − − − − − (3.3)

 

 

 

 

3.3 Life Time of Excited states:

 

As the transition probability has a dimension of inverse of time, we can find the life time of the excited state. We can thus calculate the life time of ? state ??→?+?

 

 

4.           Calculations using natural units

 

The μ meson ,or muon as it is often called, decays into three particles. These are the electron and two neutrinos one of electron type and the other of the muon type. by the inverse of ??→?+? of equation (3.34). Thus for mass of electron ? =

This is called life time of the excited state 2p.

 

4. Calculations using natural units

 

The μ meson ,or muon as it is often called, decays into three particles. These are the electron and two neutrinos one of electron type ?? and the other of the muon type ??.

If life time ? of muon decay is given by the formula

 

? ~ 2 x ??−? s. ~ 2 μs.

  1. Summary

    In this module we have applied our formalism to the hydrogen atom. We have obtained the selection rule by considering the wave functions of the atom as product of radial function and spherical harmonics. The properties of these functions give us the selection rules or permitted changes of ? and m . These selection rules are ?? =± 1 and m =0 ± 1 .

 

We studied the transition from the 2P to 1S state of hydrogen atom and calculated the transition probability per unit time. The final integral became a gamma function integral. The inverse of the transition probability gave us the life time of the excited state 2P ?. ? × ??−??. The calculation was done in the natural units.

 

We then , learnt a bit more about working in natural units and calculated the life time of muon decay.

you can view video on Application to H atom

8. Learn More Books

  1. Advanced Quantum Mechanics by J.J. Sakurai (Pearson Education Singapore (1998)
  2. Quantum Mechanics Vol.3 by L.D. Landau and Lifshitz (Pergramon Press Oxford, Reprinted
  3. “Subtle is the Lord , The science and Life of Albert Einstein” , Oxford University Press, 1982

    WEB sites

 

For history see

 

www.physics.udel.edu/~msafrono/626/Lectures%2013-14.pdf

 

www.colorado.edu/entities/quantum-field-theory/qft.history.html

 

Interesting Facts

 

P.A.M.Dirac was the first to apply quantum mechanics to absorption and emission of light in 1927. So the theory was often called Dirac theory of radiation. This was the crucial step beyond Bohr theory.