Sunday, January 30, 2011
Rotation operator on x, y in Matrix form
http://nukephysik101.wordpress.com/2011/01/30/rotation-operator-on-x-y-in-matrix-form/
Saturday, January 29, 2011
detail treatment on Larmor Precession and Rabi Resonance
http://nukephysik101.files.wordpress.com/2011/01/a-treatment-on-larmor-precession-and-rabi-resonance.pdf
is a pdf files, 15 pages. i think it covered most material on this subject. did i?
is a pdf files, 15 pages. i think it covered most material on this subject. did i?
Thursday, January 27, 2011
Announcement
the further post will post on WordPress.
since there can edit formula.
Google sucks, this time!! too bad. i am a big fan of you!
http://nukephysik101.wordpress.com/
since there can edit formula.
Google sucks, this time!! too bad. i am a big fan of you!
http://nukephysik101.wordpress.com/
Parity
It is Pa-ri-ty, not Par-ty.
Parity is just a reflection on every space dimension.
X -> -X
Y -> -Y
Z -> - Z
This is just a mirror reflection, although it only reflects on 1 dimension, the dimension that perpendicular to the mirror surface.
The parity transform is taking everything reverted. For example, when you stand up, your arms place horizontal and you left arm points forward and your right arm points right. After a parity transform. You right arm point left. Your left arm point backward, and you are standing on the ceiling, upside down. The result is a mirror image of your self. If we rotate the reverted-self from the ceiling to the ground.
Thus, parity also related as mirror reflection. In physics, we like to call the right-hand system (RHS) or left-hand system (LHS).
A simple RHS and LHS are on your hands! Although our left hand and right hand has some minor different, in general, they are the mirror image of each other. And the great interesting thing is, your left hand cannot overlap the right hand. They are equal but not the same.
Another thing is spring, when a wire is rowed clockwise and going upward, it form a left-hand spring and vice aver. Thus 2 springs are not the same.
A more physical example is the polarization of light, there are lefthand rotating light and right hand rotating light, called circular polarization. And material which interact differenty with different circular polarization are called chiral material. We should stop talking about examples in here. Because in nature, there are so many things has chiral property. Never the less, potenient and drug also has chirality. One book I recommend on general science for the chirality is "right-hand, left-hand" by chris McManus.
Physics encounters parity is because we believe if the whole world is reverted, every thing just work fine and the same. For example, if our orgasms are all reflected, left goes to right, right go to left. We still alive. In fact, there are some real cases, that some peole do have reverted orgasm. Because there should be symmetric in the world.
For those which keep function as before parity transform, we called it parity positive, for those who are not, we called it parity negative.
Be reminded that the chiral material that interact circularly polarized light different still the parity positive. For example, a material which only let right hand light passes through, but not let the left hand pass. After parity transform, it lets left hand light pass through but not right hand .Thus, the left hand and right hand are work equally well!
We also cannot say our left hand is more weak then our right hand, then we called it parity negative. It is because, if we reflected ourself, our left hand is as good and right hand and the right hand is as weak as left hand.
In normal day, parity positive never break. It is seem impossible to break. How coome some thing work differently under parity transform?
However, in mathematics, there are many parity negative things. One example is the spherical harmonic. It is can be parity positive and negative depends on the parameter.
Lets take a imaginary example in parity negative. If we use photon to hit a target, all photons are going left. Now, we reflet the whole system. But now, the photons are still going left.
The first discovery of parity negative is on beta- decay from Co-60. Whe. Applied an external magnetic field from down to up, the beta particle come out at left. When we change the magnetic field, now is from up to down, the beta particle should come out at right, if parity is positive. But it is not, it still keep coming out left!
The reason of t is beyond my understanding... Sorry.
Parity is just a reflection on every space dimension.
X -> -X
Y -> -Y
Z -> - Z
This is just a mirror reflection, although it only reflects on 1 dimension, the dimension that perpendicular to the mirror surface.
The parity transform is taking everything reverted. For example, when you stand up, your arms place horizontal and you left arm points forward and your right arm points right. After a parity transform. You right arm point left. Your left arm point backward, and you are standing on the ceiling, upside down. The result is a mirror image of your self. If we rotate the reverted-self from the ceiling to the ground.
Thus, parity also related as mirror reflection. In physics, we like to call the right-hand system (RHS) or left-hand system (LHS).
A simple RHS and LHS are on your hands! Although our left hand and right hand has some minor different, in general, they are the mirror image of each other. And the great interesting thing is, your left hand cannot overlap the right hand. They are equal but not the same.
Another thing is spring, when a wire is rowed clockwise and going upward, it form a left-hand spring and vice aver. Thus 2 springs are not the same.
A more physical example is the polarization of light, there are lefthand rotating light and right hand rotating light, called circular polarization. And material which interact differenty with different circular polarization are called chiral material. We should stop talking about examples in here. Because in nature, there are so many things has chiral property. Never the less, potenient and drug also has chirality. One book I recommend on general science for the chirality is "right-hand, left-hand" by chris McManus.
Physics encounters parity is because we believe if the whole world is reverted, every thing just work fine and the same. For example, if our orgasms are all reflected, left goes to right, right go to left. We still alive. In fact, there are some real cases, that some peole do have reverted orgasm. Because there should be symmetric in the world.
For those which keep function as before parity transform, we called it parity positive, for those who are not, we called it parity negative.
Be reminded that the chiral material that interact circularly polarized light different still the parity positive. For example, a material which only let right hand light passes through, but not let the left hand pass. After parity transform, it lets left hand light pass through but not right hand .Thus, the left hand and right hand are work equally well!
We also cannot say our left hand is more weak then our right hand, then we called it parity negative. It is because, if we reflected ourself, our left hand is as good and right hand and the right hand is as weak as left hand.
In normal day, parity positive never break. It is seem impossible to break. How coome some thing work differently under parity transform?
However, in mathematics, there are many parity negative things. One example is the spherical harmonic. It is can be parity positive and negative depends on the parameter.
Lets take a imaginary example in parity negative. If we use photon to hit a target, all photons are going left. Now, we reflet the whole system. But now, the photons are still going left.
The first discovery of parity negative is on beta- decay from Co-60. Whe. Applied an external magnetic field from down to up, the beta particle come out at left. When we change the magnetic field, now is from up to down, the beta particle should come out at right, if parity is positive. But it is not, it still keep coming out left!
The reason of t is beyond my understanding... Sorry.
on Relaxation in NMR
If we only switch on the transverse magnetic field for some time tau. after the field is off, the system will go back to the thermal equilibrium. it is due to the system is not completely isolated.
instead of consider a single spin, we have to consider the ensemble. and an ensemble is describe by the density matrix.
the reason for not consider a single spin state is, we don’t know what is going on for individual spin. in fact, in the previous section, the magnetization is a Marco effect. a single spin cannot have so many states, it can only have 2 states - up or down. if we insist the above calculation is on one spin, thus, it only give the chance for having that direction of polarization. which, is from many measurements.
so, for a single spin, the spin can only have 2 states. and if the transverse B field frequency is not equal to the Larmor frequency , and the pule is not a pi-pulse, the spin has chance to go to the other state, which probability is given by a formula. and when it goes to relax back to the minimum energy state, it will emit a photon. but when it happen, we don’t know, it is a complete random process.
However, an ensemble, a collection of spins, we can have some statistic on it. for example, the relaxation time, T1 and T2.
instead of consider a single spin, we have to consider the ensemble. and an ensemble is describe by the density matrix.
the reason for not consider a single spin state is, we don’t know what is going on for individual spin. in fact, in the previous section, the magnetization is a Marco effect. a single spin cannot have so many states, it can only have 2 states - up or down. if we insist the above calculation is on one spin, thus, it only give the chance for having that direction of polarization. which, is from many measurements.
so, for a single spin, the spin can only have 2 states. and if the transverse B field frequency is not equal to the Larmor frequency , and the pule is not a pi-pulse, the spin has chance to go to the other state, which probability is given by a formula. and when it goes to relax back to the minimum energy state, it will emit a photon. but when it happen, we don’t know, it is a complete random process.
However, an ensemble, a collection of spins, we can have some statistic on it. for example, the relaxation time, T1 and T2.
NMR
NMR is a technique to detect the state of nuclear spin. a similar technique on electron spin is call ESR ( electron spin resonance)
The principle of NMR is simple.
by analyzing the T1 and T2 and also Larmor frequency, we can known the spin, the magnetization, the structure of the sample, the chemical element, the chemical formula, and alot many others thing by different kinds of techniques.
For nuclear physics, the use of NMR is for understand the nuclear spin. for example, the polarization of the spin.
The principle of NMR is simple.
- apply a B-field, and the spin will align with it due to interaction with surrounding and precessing along the B-field with Larmor frequency. the time for the spin align with the field is call T1, longitudinal relaxation time.
- Then, we send a pule perpendicular to the B-field, it usually a radio frequency pulse. the frequency is determined by the resonance frequency, which is same as the Larmor frequency. the function of this pulse is from the B-field of it and this perpendicular B-field with perturb the spin and flip it 90 degrees.
- when the spin are rotate at 90 degrees with the static B-field, it will generate a strong enough signal around the coil. ( which is the same coil to generate the pule ) and this signal is called NMR signal.
- since the spins are not isolate, when it interact with environment, they will go back and align with the static B-field. the time for this is called T2, transverse relaxation time.
by analyzing the T1 and T2 and also Larmor frequency, we can known the spin, the magnetization, the structure of the sample, the chemical element, the chemical formula, and alot many others thing by different kinds of techniques.
For nuclear physics, the use of NMR is for understand the nuclear spin. for example, the polarization of the spin.
Wednesday, January 26, 2011
Changing of frame
sometimes, i will confuse on changing frame, especially between rotating frame and lab frame.
Tuesday, January 25, 2011
On mass deficit
The mass deficit is due to the potential energy loss.
But, why there are potential?
Coz there is a force associate with a potential.
Or in more correct way, the potential energy is due to the force. When 2 bodies in space has a attractive force, then it naturally go together. And the force accelerate the body and increase their kinetic energy.
Thus, we think the force created a field, which stored potential energy. When the force accelerates body, the potential energy converted into kinetic energy.
Imagine there are 2 bodies, A & B, Which at different location and have different force. If A is further away, when it comes at B, it has more kinetic energy than B, coz B is at rest at that position. Thus, further away, higher the potential.
Can thus potential be as high as infinite? In reality, because thing has surface, has size. The minimum distance between the 2 bodies is the sum of their radius. And if we sum up all potential, from the surface to infinity. We found that it is finite. And we like to set the potential at infinity is zero. Thus, it makes all attractive force has negative potential.
But, electric and positron do not has size, so, and electron and a positron can come together and the potential energy they can release is infinite! Since when they are apart infinite and the attractive force accelerate it, when they meet, their speed will be infinite!
Wait! The speed cannot be infinite, the highest speed is the speed of light and for an object has mass, it never move at the speed of light!
So, when electron and positron hit each other from infinity. However, relativity does not limit the energy and the kinetic energy. The result is, it can release infinite energy.
But at first, infinite exist. In reality, we can just approximate the "infinite" by far far away. When the potential change very little. But far far away is still very short compare to infinite. Thus, we don't have infinite energy source.
In laboratory, we can accelerate electron and position at very high speed and contain several GeV. Thus when those particles annihilate, they release GeV energy due to the kinetic energy and their mass, which is just 0.000511GeV. Does not help much.
Monday, January 24, 2011
Mass of particles and nucleus
in Nuclear physics, the particle we deal with are so small and so light, if we use standard unit, then there will be many zero and we will lost in the zeros. for example, the electron has mass:
Mass( electron ) = 9.11 × 10-31 kilograms
Mass( proton ) = 1.67 × 10-27 kilograms
see? as the special relativity give us a translation tool - E = m c^2, thus, we can use MeV to talk about mass.
Mass ( electron ) = 0.511 MeV
Mass ( proton ) = 938.3 MeV
thus, we can see, Proton is roughtly 2000 times heavier then electron ( 1000 : 0.5 ).
Mass( neutron ) = 939.6 MeV
neutron is just 1.3 MeV heavier then proton.
The nucleus is formed by proton and neutron. so, in simple thought, an nucleus with Z proton and ( A-Z ) neutron should have mass
Z x Mass( proton ) + ( A - Z ) x Mass ( neutron ) = Mass ( A, Z )
where A is the atomic mass number, which is equal the number of nucleons in the nucleus, and Z is the proton number.
However, scientists found that it is not true.
Z x Mass( proton ) + ( A - Z ) x Mass ( neutron ) > Mass ( A, Z )
Some of the mass is missing! But that is explained why nucleus will not break down automatically. since it need extra energy to break it down.
we called the mass different is Mass Deficit. or Blinding energy.
Mass Deficit = Mass( A, Z) - Mass ( proton + neutron )
some one may think that the blinding energy is the energy for holding the nucleus together. in order to hold the nucleus, some mass was converted into the energy to holding it. this is INCORRECT. the correct argument is, the blinding energy is th energy require to break it down.
think about a simple 2 bodies system, like sun and earth. at far far away, when both of them are at rest, the total energy is Mass( sun) + Mass ( earth ) + Potential energy.
when the earth moves toward to sun, the potential energy converted to the Kinetic energy, so the earth moving faster and faster. but, in order to stay in the orbit, some K.E. must be lost so that it does not have enough ( or the same) energy to run away. Thus, the total energy of the system is lesser then the total mass.
another analogy is electron orbit. when an electron was captured by an atom, it radiate energy in order to stay in some energy level. thus, the total energy of the system again less then the total mass.
any any case, the mass of the sun and earth and electron does not change, but the potential changes to negative, thus it makes to total energy lesser.
similar idea hold for nucleus, but the potential of it are great different, because there are a Coulomb Barrier. Thus, in order to make a nucleus. we have to put so many K.E. to again this barrier, then the resultant nucleus release the Mass Deficit energy and also the input K.E..
Mass( electron ) = 9.11 × 10-31 kilograms
Mass( proton ) = 1.67 × 10-27 kilograms
see? as the special relativity give us a translation tool - E = m c^2, thus, we can use MeV to talk about mass.
Mass ( electron ) = 0.511 MeV
Mass ( proton ) = 938.3 MeV
thus, we can see, Proton is roughtly 2000 times heavier then electron ( 1000 : 0.5 ).
Mass( neutron ) = 939.6 MeV
neutron is just 1.3 MeV heavier then proton.
The nucleus is formed by proton and neutron. so, in simple thought, an nucleus with Z proton and ( A-Z ) neutron should have mass
Z x Mass( proton ) + ( A - Z ) x Mass ( neutron ) = Mass ( A, Z )
where A is the atomic mass number, which is equal the number of nucleons in the nucleus, and Z is the proton number.
However, scientists found that it is not true.
Z x Mass( proton ) + ( A - Z ) x Mass ( neutron ) > Mass ( A, Z )
Some of the mass is missing! But that is explained why nucleus will not break down automatically. since it need extra energy to break it down.
we called the mass different is Mass Deficit. or Blinding energy.
Mass Deficit = Mass( A, Z) - Mass ( proton + neutron )
some one may think that the blinding energy is the energy for holding the nucleus together. in order to hold the nucleus, some mass was converted into the energy to holding it. this is INCORRECT. the correct argument is, the blinding energy is th energy require to break it down.
think about a simple 2 bodies system, like sun and earth. at far far away, when both of them are at rest, the total energy is Mass( sun) + Mass ( earth ) + Potential energy.
when the earth moves toward to sun, the potential energy converted to the Kinetic energy, so the earth moving faster and faster. but, in order to stay in the orbit, some K.E. must be lost so that it does not have enough ( or the same) energy to run away. Thus, the total energy of the system is lesser then the total mass.
another analogy is electron orbit. when an electron was captured by an atom, it radiate energy in order to stay in some energy level. thus, the total energy of the system again less then the total mass.
any any case, the mass of the sun and earth and electron does not change, but the potential changes to negative, thus it makes to total energy lesser.
similar idea hold for nucleus, but the potential of it are great different, because there are a Coulomb Barrier. Thus, in order to make a nucleus. we have to put so many K.E. to again this barrier, then the resultant nucleus release the Mass Deficit energy and also the input K.E..
 |
| a scratch on the nuclear potential. there are a Coulomb Barrie. ( by wolframalpha.com) |
When the nucleus is radioactive and undergoes decay. this mean, it Mass deficit is positive. thus, it will automatically break down to another nucleus until it mass deficit is negative again. during this process, the emitted particle carry K.E. which is from the potential. Not the mass for one nucleons.
Remember, Mass( nucleus ) = Mass ( protons + neutrons ) + Potential
Larmor Procession
Magnetic moment (mu) :
this is a magnet by angular momentum of charge or spin. its value is:
mu = gamma S ( in case of spin ) = gamma L ( in case of angular momentum )
where gamma is (g-factor) x ( Bohr magneton ) / ( h_bar [ in case that the S or L is not using natural unit ] )
the g-factor is a dimensionless number, which reflect the environment of the spin, for orbital angular momentum, g = 1.
Bohr magneton = electron charge x h_bar / ( 2 mass), since different particle has different mass, their Bohr magneton value are different. electron is the lightest particle, so, it has largest value on Bohr magneton.
J-coupling ( i am not sure it is correct to say so)
since the spin and angular momentum are mathematical equal. they give the same effect when react with magnetic field, thus, we use a new term:
J = L + S
mu = gamma J
in this case, since the g-factor of L and S are different, and when we combine with different particles, the Bohr magnetons are different, in that case, special treatment is needed. but it does not change the principle.
Larmor frequency:
When applied a magnetic field on a magnetic moment, the field will cause the moment process around the axis of the field. the precession frequency is called Larmor frequency.
the procession can be understood in classical way or QM way.
Classical way:
the change of angular momentum is equal to the applied torque. and the torque is equal to the magnetic moment cross product with the magnetic field.
d J / d t = mu x B = gamma J x B
solving gives the procession frequency is :
w = - gamma B
the minus sign is very important, it indicated that the J is precessing by left hand rule.
QM way:
The Schrödinger equation is :
i d/dt |Psi > = H |Psi >
H is the Hamiltonian = - mu . B = - gamma J.B = - gamma B Jz = w Jz
Jz is an operator.
the solution is for |Psi> is
|Psi ( t ) > = Exp[ - i w Jz ] |Psi ( t = 0 ) >
for up state coefficient a1 and down state coefficient a2,
solving a1 and a2 gives:
a1 = Exp [ - i w t ]
a2 = Exp [ + i w t ]
Thus, in QM point of view, the state does not change but only the phase.
For not using natural unit, we knew, w h_bar = Energy.
this is a magnet by angular momentum of charge or spin. its value is:
mu = gamma S ( in case of spin ) = gamma L ( in case of angular momentum )
where gamma is (g-factor) x ( Bohr magneton ) / ( h_bar [ in case that the S or L is not using natural unit ] )
the g-factor is a dimensionless number, which reflect the environment of the spin, for orbital angular momentum, g = 1.
Bohr magneton = electron charge x h_bar / ( 2 mass), since different particle has different mass, their Bohr magneton value are different. electron is the lightest particle, so, it has largest value on Bohr magneton.
J-coupling ( i am not sure it is correct to say so)
since the spin and angular momentum are mathematical equal. they give the same effect when react with magnetic field, thus, we use a new term:
J = L + S
mu = gamma J
in this case, since the g-factor of L and S are different, and when we combine with different particles, the Bohr magnetons are different, in that case, special treatment is needed. but it does not change the principle.
Larmor frequency:
When applied a magnetic field on a magnetic moment, the field will cause the moment process around the axis of the field. the precession frequency is called Larmor frequency.
the procession can be understood in classical way or QM way.
Classical way:
the change of angular momentum is equal to the applied torque. and the torque is equal to the magnetic moment cross product with the magnetic field.
d J / d t = mu x B = gamma J x B
solving gives the procession frequency is :
w = - gamma B
the minus sign is very important, it indicated that the J is precessing by left hand rule.
QM way:
The Schrödinger equation is :
i d/dt |Psi > = H |Psi >
H is the Hamiltonian = - mu . B = - gamma J.B = - gamma B Jz = w Jz
Jz is an operator.
the solution is for |Psi> is
|Psi ( t ) > = Exp[ - i w Jz ] |Psi ( t = 0 ) >
for up state coefficient a1 and down state coefficient a2,
solving a1 and a2 gives:
a1 = Exp [ - i w t ]
a2 = Exp [ + i w t ]
Thus, in QM point of view, the state does not change but only the phase.
For not using natural unit, we knew, w h_bar = Energy.
Natural unit
on Size and Unit, we know that the speed of light better be equal to 1, that simplify the equation of relativity.
now, we impose 1 more things, the Reduced Planck constant, h_bar also set to 1. that simplify all equations with angular momentum or spin.
the Angular momentum:
L^2 | l, m > = h_bar l ( l+1) |l, m >
Lz |l, m > = h_bar m |l, m >
now becomes :
now, we impose 1 more things, the Reduced Planck constant, h_bar also set to 1. that simplify all equations with angular momentum or spin.
the Angular momentum:
L^2 | l, m > = h_bar l ( l+1) |l, m >
Lz |l, m > = h_bar m |l, m >
now becomes :
L^2 | l, m > = l ( l+1) |l, m >
Lz |l, m > = m |l, m >
this is same for the spin.
by doing so, the magnetic moment will be
mu = g x muB x S or g x muB x L
since there is no h_bar in S or L anymore.
Saturday, January 22, 2011
Spin
( this is just a draft, not organized )
Spin is a intrinsics property of elementary particle, such as electron, proton, and even photon. Intrinsics means it is a built-in property, like mass, charge. Which extrinsic properties are speed, momentum.
Spin is a vector or tensor quality while charge and mass are scaler.
Spin can react with magnetic field, like charge reacts with electric field or mass react with force produce acceleration. Thus, spin is like a bar-magnet inside particle, counter part of charge.
The magnitude of spin is half integer or integer of reduced Planck's constant h_bar or hb. Particles with half integer of spin are classified as Fermion, and those with integer spin are Boson. they follow different statistic while interact together, thus, this creates different physics for different group.
we are not going to the mathematic description this time.
the effect of spin causes the magnetic moment, that's why it react with magnetic field. the other thing that creates magnetic moment is angular momentum for charge particle, like electron orbiting around nucleus. So, both spin and angular momentum can be imagined as a little magnet, thus, they can interact, in physics, we call the interaction between spin and angular momentum is coupling. for example, spin-orbital coupling, spin-spin coupling, etc..
when the spin interact with external magnetic field, it will process around the magnetic field with Larmor frequency. and the direction of the spin while undergoes procession can only be certain angle. for spin half, like electron or proton. there are only 2 directions, and we called it up and down.
Spin is a intrinsics property of elementary particle, such as electron, proton, and even photon. Intrinsics means it is a built-in property, like mass, charge. Which extrinsic properties are speed, momentum.
Spin is a vector or tensor quality while charge and mass are scaler.
Spin can react with magnetic field, like charge reacts with electric field or mass react with force produce acceleration. Thus, spin is like a bar-magnet inside particle, counter part of charge.
The magnitude of spin is half integer or integer of reduced Planck's constant h_bar or hb. Particles with half integer of spin are classified as Fermion, and those with integer spin are Boson. they follow different statistic while interact together, thus, this creates different physics for different group.
we are not going to the mathematic description this time.
the effect of spin causes the magnetic moment, that's why it react with magnetic field. the other thing that creates magnetic moment is angular momentum for charge particle, like electron orbiting around nucleus. So, both spin and angular momentum can be imagined as a little magnet, thus, they can interact, in physics, we call the interaction between spin and angular momentum is coupling. for example, spin-orbital coupling, spin-spin coupling, etc..
when the spin interact with external magnetic field, it will process around the magnetic field with Larmor frequency. and the direction of the spin while undergoes procession can only be certain angle. for spin half, like electron or proton. there are only 2 directions, and we called it up and down.
Wednesday, January 19, 2011
Hall effect
It is a short review.
The hall probe is perpendicular to the B field( pointing up) and have a current I passing through ( going forward ).
Due to the Lorentz force. The positron is moving to right and accumulate. The accumulating charge creates a electric force to the left to against further positron accumulate. The magnetic force will be balanced by the electric force. Due to the electric force, there is associated voltage across the hall probe. This voltage is called hall voltage.
FB= e v B = Vh d
Where e is positron charge, v is speed of positron, B is the magnetic field, Vh is the Hall voltage and d is the distance across the hall probe.
The current I is
I = A n e v
Where A is the cross section area of the hall probe, n is density of the positron carrier, v is the positron velocity.
Thus,
Vh = I B / ( V n e )
Where V is the volume of the hall probe. But the V n is equal to the total number of positron N.
Vh = I B / ( N e )
Which is to say, the hall voltage is proportional to the magnetic field.
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.
**********************************
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.
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.
**********************************
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.
WKB approximation
I was scared by this term once before. in fact, don't panic, it is easy. Let me explain.
i just copy what written in Introduction to Quantum Mechanics by David Griffiths (1995) Chapter 8.
The approx. can be applied when the potential is varies slowly compare the wavelength of the wave function. when it expressed in Exp[ i k x], wavelength = 2 pi / k, when it expressed in Exp[ - kapper x ], wavelength = 1/kapper.
in general, the wavefunction can be expressed as amplitude and phase:
A[x] Exp[ i phase[x] ], A[x] and phase[x] are real function.
sub this into the Schrödinger equation ( Psi ''[x] = - { 2 m / hb (E - V[x])} Psi[x] ). and separate the imaginary part and real part.
The imaginary part is can be simplified as:
D[ A^2 phase' ] = 0 => A = Const. /Sqrt[ phase'[x] ]
The real part is
A''[x] = ( phase'[x]^2 - 2 m / hb (E - V[x]) ) A[x]
we use the approx. that A''[x] = 0 since it varies slowly.
Thus,
phase'[x] = Sqrt[2 m / hb (E - V[x])]
=>
phase[x] = Integrate[ Sqrt[2 m / hb (E - V[x])], x]
and the solution is, ( if we use p[x] = Sqrt[2 m / hb (E - V[x])] )
Psi [x] = Const. /Sqrt[ p[x] ] Exp[ i Integrate[ p[x] ] ]
Simple! but one thing should keep in mind that, the WKB approx is not OK when Energy = potential.
This tell you, the phase part of the wave function is equal the square of the area of the different of Energy and the Potential.
when the energy is smaller then the potential, than, the wavefunction is under decay.
one direct application of WKB approxi is on the Tunneling effect.
if the potential is large enough, so, the transmittance is dominated by the decay, Thus, the probability of the tunneling is equal to Exp[- 2 Sqrt[ 2m / hb] Sqrt[area between V[x] and Energy] ]. Therefore, when we have an ugly potential, we can approx it by a rectangular potential with same area to give the similar estimation.
i just copy what written in Introduction to Quantum Mechanics by David Griffiths (1995) Chapter 8.
The approx. can be applied when the potential is varies slowly compare the wavelength of the wave function. when it expressed in Exp[ i k x], wavelength = 2 pi / k, when it expressed in Exp[ - kapper x ], wavelength = 1/kapper.
in general, the wavefunction can be expressed as amplitude and phase:
A[x] Exp[ i phase[x] ], A[x] and phase[x] are real function.
sub this into the Schrödinger equation ( Psi ''[x] = - { 2 m / hb (E - V[x])} Psi[x] ). and separate the imaginary part and real part.
The imaginary part is can be simplified as:
D[ A^2 phase' ] = 0 => A = Const. /Sqrt[ phase'[x] ]
The real part is
A''[x] = ( phase'[x]^2 - 2 m / hb (E - V[x]) ) A[x]
we use the approx. that A''[x] = 0 since it varies slowly.
Thus,
phase'[x] = Sqrt[2 m / hb (E - V[x])]
=>
phase[x] = Integrate[ Sqrt[2 m / hb (E - V[x])], x]
and the solution is, ( if we use p[x] = Sqrt[2 m / hb (E - V[x])] )
Psi [x] = Const. /Sqrt[ p[x] ] Exp[ i Integrate[ p[x] ] ]
Simple! but one thing should keep in mind that, the WKB approx is not OK when Energy = potential.
This tell you, the phase part of the wave function is equal the square of the area of the different of Energy and the Potential.
when the energy is smaller then the potential, than, the wavefunction is under decay.
one direct application of WKB approxi is on the Tunneling effect.
if the potential is large enough, so, the transmittance is dominated by the decay, Thus, the probability of the tunneling is equal to Exp[- 2 Sqrt[ 2m / hb] Sqrt[area between V[x] and Energy] ]. Therefore, when we have an ugly potential, we can approx it by a rectangular potential with same area to give the similar estimation.
alpha decay
different decay cause by deferent mechanism, we first start on alpha decay.
i assume we know what is alpha decay, which is a process that bring excited nucleus to lower energy state by emitting an alpha particle.
The force govern this process is the strong force, due to the force is so strong, the interaction time is very short, base on the uncertainty principle that large change in energy leads to short time interval. however, the observed alpha decay constant is about 1.3 × 1010 year, which is about the age of our universe. That's why we still able to find it at the beginning of nuclear physics : discovery of radioactive matter.
The reason for such a long decay time is due to the Coulomb barrier of the nuclear potential. since the proton carry positive charge, thus. it creates a positive potential wall in the nucleus. that potential not only repulse proton from outside but also the proton from inside which try to get out. thus, the inside protons are bounded back and forth inside the nucleus. due to the momentum carried by the protons, it has frequency 6 × 1021 per sec.
Due to the Quantum tunneling effect, the probability of tunneling is 4 × 10-40. which is a very small chance. But , don't forget there are 6 × 1021 trails per sec. Thus, the chance per sec is 2.4 × 10-18 . and the mean life time is inverse of the probability, thus it is approx 1.3 × 1010 year.
i assume we know what is alpha decay, which is a process that bring excited nucleus to lower energy state by emitting an alpha particle.
The force govern this process is the strong force, due to the force is so strong, the interaction time is very short, base on the uncertainty principle that large change in energy leads to short time interval. however, the observed alpha decay constant is about 1.3 × 1010 year, which is about the age of our universe. That's why we still able to find it at the beginning of nuclear physics : discovery of radioactive matter.
The reason for such a long decay time is due to the Coulomb barrier of the nuclear potential. since the proton carry positive charge, thus. it creates a positive potential wall in the nucleus. that potential not only repulse proton from outside but also the proton from inside which try to get out. thus, the inside protons are bounded back and forth inside the nucleus. due to the momentum carried by the protons, it has frequency 6 × 1021 per sec.
Due to the Quantum tunneling effect, the probability of tunneling is 4 × 10-40. which is a very small chance. But , don't forget there are 6 × 1021 trails per sec. Thus, the chance per sec is 2.4 × 10-18 . and the mean life time is inverse of the probability, thus it is approx 1.3 × 1010 year.
Tuesday, January 18, 2011
terminology
this part is for people who confess and poor in english :P
Nucleus : (noun) is the core of element, which contain protons and neutrons
Nuclei : (noun) many of nucleus
Nuclear : (adj) to relate something to nucleus. e.g. nuclear energy, nuclear plant, nuclear phsyics
Nucleon: (noun) is the thing make the nucleus, which is a single word. a nucleon can be a proton or neutron
Nuclide: ( noun) [from google translate : A distinct kind of atom or nucleus characterized by a specific number of protons and neutrons ] (what??)
Nucleus : (noun) is the core of element, which contain protons and neutrons
Nuclei : (noun) many of nucleus
Nuclear : (adj) to relate something to nucleus. e.g. nuclear energy, nuclear plant, nuclear phsyics
Nucleon: (noun) is the thing make the nucleus, which is a single word. a nucleon can be a proton or neutron
Nuclide: ( noun) [from google translate : A distinct kind of atom or nucleus characterized by a specific number of protons and neutrons ] (what??)
Detector
there are many types of detectors developed. now is mostly electronic and able to collect huge amount of data.
there are 3 main principles on particle detection.
here is a diagram from WIKI, which summarized
there are 3 main principles on particle detection.
- by the ionization between particle and gas
- by the excitation of energy level
- by the detection of photon
i will talk more about them when i feel i understand. :)
here is a diagram from WIKI, which summarized
on the sum of 4 momentum and excited mass
when we have a decay process, there are many fragments, we can measure their momentum and energy and construct the 4-momentum
P_i = ( E_i , p_i )
we use the c = 1 unit as usual.
to find out the mass before the decay, we can use
Sqrt[ Sum[E_i]^2 - Sum[p_i]^2 ] = excited mass.
the reason for the term "excited mass", we can see by the following illustration.
consider a head on collision of 2 particles in C.M. frame, with momentum p and energy E1 and E2.
the mass for each one is given by
m1=Sqrt[ E1^2 - p^2 ]
m2=Sqrt[ E2^2 - p^2 ]
but if we use the sum of the 4 momentum and calculate the mass,
Sqrt[ (E1+E2) ^2 - (p - p)^2 ] = E1 + E2
which is not equal to Sqrt[ E1^2 - p^2 ] + Sqrt[ E2^2 - p^2 ]
in fact, it is larger.
the reason for its larger is, when using the sum of 4 momentum, we actually assumed the produce of collision is just 1 particles, and the collision is inelastic. Thus, if we think about the time-reverse process, which is a decay, thus, some of the mass will convert to K.E. for the decay product.
P_i = ( E_i , p_i )
we use the c = 1 unit as usual.
to find out the mass before the decay, we can use
Sqrt[ Sum[E_i]^2 - Sum[p_i]^2 ] = excited mass.
the reason for the term "excited mass", we can see by the following illustration.
consider a head on collision of 2 particles in C.M. frame, with momentum p and energy E1 and E2.
the mass for each one is given by
m1=Sqrt[ E1^2 - p^2 ]
m2=Sqrt[ E2^2 - p^2 ]
but if we use the sum of the 4 momentum and calculate the mass,
Sqrt[ (E1+E2) ^2 - (p - p)^2 ] = E1 + E2
which is not equal to Sqrt[ E1^2 - p^2 ] + Sqrt[ E2^2 - p^2 ]
in fact, it is larger.
the reason for its larger is, when using the sum of 4 momentum, we actually assumed the produce of collision is just 1 particles, and the collision is inelastic. Thus, if we think about the time-reverse process, which is a decay, thus, some of the mass will convert to K.E. for the decay product.
Friday, January 14, 2011
Type of accelerator (Ring type)
Ring type accelerator solved the difficulties by linear type.
- the particle circulate inside the accelerator, so, it can be accelerate infinite time in principle.
- The space require is smaller compare with similar energy output Linac.
However, there is a draw back is, for charged particle running in a circular path, it will radiate energy by EM wave due to the centripetal acceleration, thus, even it is just running in constant speed, it will radiate and energy lost. this is called synchrotron radiation.
There are mainly 2 types of ring accelerators, 1) cyclotron, 2) synchrotron.
Cyclotron
cyclotron is the simplest type, it has 2 D shape cavities and the 2 D shape formed a circle. the 2 D shape cavities is under a magnetic field to blend the particle. and the 2 Ds have different electric potential. when a particle pass from 1 D to the others, due to the potential different, it will be accelerate.
as you can imagine, the potential of the Ds has to be oscillating so that the particle is accelerated when passing each gap between the 2Ds. That frequency is called cyclotron frequency. and it also reflects the particle circulating frequency. surprisingly, the cyclotron frequency only depends on the magnetic field strength, the charge and the mass of the particle.
f = (B q) / (2 pi m)
which means, no matter the particle position, it moves in same frequency. Thus, the outer particle move faster then the inner one.
now a day, cyclotron may not just contains 2 D cavities but any 2 pi/n cavities. where n is number of cavities. thus, 2Ds is also called pi - cavity.
The typical speed it can reach is about 10% of speed of light.
The only draw back is, the energy it can reach is limited, if using fixed B field or E field, due to relativistic effect. (i.e. the cyclotron frequency also depend on the speed ) the particle cannot match the frequency and accelerated, after it goes to relativistic speed.
another factor is the B field strength is limited, even using super conductive magnetic. and the limit of B field, limited the max output.
particle is released at the center of the B-field and go outward as it acquire speed. Thus, the limitation of radius also limited the max speed. and also, a large radius means a large B field area, which raise a problem on uniform on the B-field.
So, there are another type of cyclotron, which changing the B field or E field to cope with the changing frequency. such cyclotron is called Synchrocyclotron. but due to the velocity dependent of the frequency, only certain speed of particle can be accelerate, thus, the intensity of the beam is smaller then cyclotron.
Synchrotron
synchrotron can reach a great energy and accelerate particle very close to speed of light.
it uses a lot beam focusing devices and accelerating devices to accelerate the beam in a very large radius. each device is well tuned, and all devices are well synchronized for different particle. Thus it is a very delicates and sophisticated machine.
- particle can have every high energy
- high intensity of beam
- it can have some section only for linear motion with accelerate.
- it is not limited by the B field. since the narrow of the beam, a higher forcing B field can be applied.
The only factor reduced the power output is the synchrotron radiation.
particles identification
after a bombardment in accelerator, there are many fragments, different kind of particles produced. thus, the identification of particles (PID) is essential.
there are 3 main principles in PID.
there are 3 main principles in PID.
- decay mood
- time of flight
- energy loss to surrounding
Monday, January 10, 2011
Scattering phase shift
for a central potential, the angular momentum is a conserved quantity. Thus, we can expand the wave function by the angular momentum wave function:
Sum { a[ l ] Y[ l, m = 0] R[ l , k , r] }
the m=0 is because the spherical symmetry. the R is the radial part of the wave function. and a is a constant. k is the linear momentum and r is the radial distance.
for free particle, potential equal to zero,
R -> J[ l , k r]
which is reasonable when r is infinite and the nuclear potential is very short distance. when r goes to infinity,
J[ l, k r ] -> 1/ (k r ) Sin{ k r - 1/2 l pi}
for elastic scattering, the probability of the current density is conserved in each angular wave function, thus,
the effect of the nuclear potential can only change the phase inside the sin function:
1/ (k r ) Sin{ k r - 1/2 l pi + d[ l ]}
with further treatment, the total cross section is proportional to Sin{ d[ l ] }^2.
thus, by knowing the scattering phase shift, we can know the properties of the nuclear potential.
for more detail : check this website
Sum { a[ l ] Y[ l, m = 0] R[ l , k , r] }
the m=0 is because the spherical symmetry. the R is the radial part of the wave function. and a is a constant. k is the linear momentum and r is the radial distance.
for free particle, potential equal to zero,
R -> J[ l , k r]
which is reasonable when r is infinite and the nuclear potential is very short distance. when r goes to infinity,
J[ l, k r ] -> 1/ (k r ) Sin{ k r - 1/2 l pi}
for elastic scattering, the probability of the current density is conserved in each angular wave function, thus,
the effect of the nuclear potential can only change the phase inside the sin function:
1/ (k r ) Sin{ k r - 1/2 l pi + d[ l ]}
with further treatment, the total cross section is proportional to Sin{ d[ l ] }^2.
thus, by knowing the scattering phase shift, we can know the properties of the nuclear potential.
for more detail : check this website
Sunday, January 9, 2011
Informations we can extract
in scattering experiment, the raw informations we can know or observe are only 2 things:
- the number of particles counted at particular solid angle. ( when you have a unit sphere, the area on the surface is called solid angle)
- The polarization (spin)
- charge
- energy
- momentum (time of flight)
Since the number of particles counted is related to the intensity of the incident beam, the density of the target, the interaction and the differential cross section.
on the other hand, the number of particles counted should be related to intensity of incident beam, density of the target and interaction potential. Thus, the differential cross section is related to the interaction potential.
The polarization can be measured by 2nd scattering of known polarization target. or directly from a polarized primary target.
magic number
we knew that for some atoms are more stable that others. like He, Ne, Ar, etc, which are belonged to noble gas. the reason for they are un-reactive is, there outer most electron shell is filled out.
similar things happened in nuclei. in the shell model of nuclei, protons and neutrons just like the electrons in atom. if the outer most shell of proton or neutron is filled out, the nuclei is very stable. and we called this number of proton or neutron be MAGIC NUMBER.
the first magic number is 2. this nuclei of 2 protons is very stable. If there are 2 protons and 2 neutrons, we called this double magic number, and this nuclei, which is He is very very stable.
the list of magic number is 2, 8, 20, 28, 50, 82, 126 in theory prediction.
however, when the nuclei become heavier and heavier, the stability of nuclei in the magic number lost. to understand this. we have to know that the magic number is come from the large spin-orbital coupling term in the Hamiltonian of the nuclei. and recent research suggest that, the spin-orbital coupling may change by the number of nucleons.
Saturday, January 8, 2011
Thursday, January 6, 2011
Slide rule
um... i don't think it is good to post in here.
moved to : http://gucciproxy.blogspot.com/2011/01/slide-rule.html
and http://gucciproxy.blogspot.com/2011/01/slide-rule-2.html
moved to : http://gucciproxy.blogspot.com/2011/01/slide-rule.html
and http://gucciproxy.blogspot.com/2011/01/slide-rule-2.html
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