decay time constant and line width

the spectrum of energy always has a peak and a line width.

the reason for the line width is, this is decay.

i give 2 explanations, once is from classical point of view and i skipped the explanation for the imaginary part. so, i am not fully understand. the 2nd explanation is look better, but it is from QM. however, there is one hide question for that explanation is, why the imaginary energy is negative?

the simplest understanding of the relation is using fourier transform. (i think)

fourier transform is changing the time-frame into the frequency frame. i.e, i have a wave, propagating with frequency w. we can see a wave shape when plot with time. and we only see a line, when we plot with frequency, since there is only 1 single frequency. however, for a general wave, it is composite of many different frequencies, using fourier transform can tell us which frequency are involved. And energy is proportional to frequency.

when the particle or state under decay. the function is like

Exp[- R t] Exp[ i wo t]

where the R is decay constant, and wo is the wave frequency.

after fourier transform, assume there is nothing for t < 0

1/ ( R + i ( wo - w) )

the real part is

R/ ( R^2 + (w - wo)^2 )

which is a Lorentzian shape and have Full-Width-Half-Maximum (FWHM) is 2R. it comes from the cosine part of the fourier transform. thus, the real part.

and the imaginary part is

(w - wo)/( R^2 + (w - wo)^2 )

the imaginary part is corresponding to the since part, so, we can neglect it. (how exactly why we can neglect it? )

Thus, we can see, if there is no decay, R -> 0, thus, there is no line width.

therefore, we can see the line width in atomic transition, say, 2p to 1s. but there are many other mechanism to the line width, like Doppler broadening, or power broadening. So, Decay will product line width, but not every line width is from decay.

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another view of this relation is from the quantum mechanics.

the solution of Schroedinger equation is

Psi [x, t ] = phi[x] Exp[- i E/hbar t]

so, the probability conserved with time, i.e.:

| Psi [x,t] |^2 = | Psi[x,0] | ^2

if we assume the energy has small imaginary part

E = E0 - i R hbar/2 ( why the imaginary energy is nagative?)

 | Psi [x,t] |^2 = | Psi[x,0] | ^2 Exp[- R t]

that make the wavefunction be :

Psi [x, t ] = phi[x] Exp[- i E/hbar t] Exp[ - R/2 t ]

what is the meaning of the imaginary energy?

the wave function is on time-domain, but what is "physical", or observable is in Energy -domain. so, we want Psi[x,E] rather then Psi[x,t], the way to do the transform is by fourier transform.

and after the transform, the probability of finding particle at energy E is given by

|Psi[x,E]|^2 = Constant / ( R^2 + ( wo - w)^2 )

which give out the line width in energy.

and the relation between the FWHM(line width) and the decay time is

mean life time = hbar / FWHM

which once again verify the uncertainty principle.

Hydrogen Atom (Bohr Model)

OK, here is a little off track. But that is what i were learning and learned. like to share in here. and understand the concept of hydrogen is very helpful to understand the nuclear, because many ideas in nuclear physics are borrow from it, like "shell".

The interesting thing is about the energy level of Hydrogen atom. the most simple atomic system. it only contains a proton at the center, um.. almost center, and an electron moving around. well, this is the "picture". the fact is, there is no "trajectory" or locus for the electron, so technically, it is hard to say it is moving!

why i suddenly do that is because, many text books said it is easy to calculate the energy level and spectrum for it. Moreover, many famous physicists said it is easy. like Feynman, Dirac, Landau, Pauli, etc... OK, lets check how easy it is.

anyway, we follow the usual way in every text book. we put the Coulomb potential in the Schrödinger equation, change the coordinate to spherical. that is better and easy for calculation because the coulomb potential is spherical symmetric. by that mean, the momentum operator (any one don't know what is OPERATOR, the simplest explanation is : it is a function of function.) automatically separated into 2 parts : radial and angular part. The angular part can be so simple that it is the Spherical harmonic.

Thus the solution of the "wave function" of the electron, which is also the probability distribution of  the electron location, contains 2 parts as well. the radial part is not so trivial, but the angular part is so easy. and it is just Y[l,m].

if we denote the angular momentum as L, and the z component of it is Lz, thus we have,

L²Y[l,m] = l (l+1) Y[l,m]
Lz Y[l,m] = m Y[l,m]

as every quadratic operator, there are "ladder" operator for "up" and "down".

L⁺ Y[l,m] =const. Y[l,m+1]
L^- Y[l,m] = const. Y[l,m-1]

which means, the UP operator is increase the z-component by 1, the constant there does not brother us.

it is truly easy to find out the exact form of the Y[l,m] by using the ladder operator. as we know, The z component of the a VECTOR must have some maximum. so, there exist an Y[l,max] such that

L⁺ Y[l,max] = 0

since there is no more higher z-component.

by solve this equation, we can find out the exact form of Y[l,max] and sub this in to L², we can know max = l. and apply the DOWN operator, we can fins out all Y[l,m], and the normalization constant is easy to find by the normalization condition in spherical coordinate, the normalization factor is sin[theta], instead of 1 in rectangular coordinate.