Arthur Compton and Debye both provided in 1922 a very simple mathematical framework for the momentum of these photons with Compton having experimental evidence from firing X-Rays of known frequency into graphite and looking at recoil electrons.
Let E = mc² = hf for a photon, where f is frequency, and "m" is the mass "equivalent" of the photon given they have no "rest mass". (It is important to recognise that stopping a photon to measure its mass eliminates it -so it has no "at rest" mass - crucial in Special Relativity where, to travel at the speed of light, mass would otherwise become infinite.)
Having "rigged" this mass problem,
p = momentum = mc (mass x velocity) = hf/c = E/c = h/l
The experiment shows that X-Rays and electrons behave exactly like ball bearings colliding on a table top using the same 2D vector diagrams. They enter the graphite at one wavelength and leave at a longer wavelength as they have transfered both momentum and kinetic energy to an electron. Momentum and energy are conserved in the collision if we accept the equation above for momentum of light.
When the photon enters at l0 and leaves at l1, its energy has changed from E0 to E1 and momentum from E0/c to E1/c with a change in direction of q. The electron gains Ek = E0 - E1
See for a diagram and more details
http://www.launc.tased.edu.au/online/sciences/physics/compton.html
The Compton Effect ( a different explanation)
Convincing evidence that light is made up of particles (photons), and that photons have momentum, can be seen when a photon with energy hf collides with a stationary electron. Some of the energy and momentum is transferred to the electron (this is known as the Compton effect), but both energy and momentum are conserved in this elastic collision. After the collision the photon has energy hf' and the electron has acquired a kinetic energy K.
Conservation of energy: hf = hf' + K
Combining this with the momentum conservation equations, it can be shown that the wavelength of the outgoing photon is related to the wavelength of the incident photon by the equation:
Δλ = λ' - λ = (h/mec)(1 - cosq)
The combination of factors h/mec = 2.43 x 10-12 m, where me is the mass of the electron, is known as the Compton wavelength. The collision causes the photon wavelength to increase by somewhere between 0 (for a scattering angle of 0°) and twice the Compton wavelength (for a scattering angle of 180°).
Source:
http://physics.bu.edu/~duffy/semester2/c35_compton.html
It has problems and examples and demonstration also.
Tuesday, February 10, 2009
Saturday, February 7, 2009
Schrodinger’s Equation
Quantum mechanics describes the spectra in a much better way than Bohr’s model.
Electron has a wave character as well as a particle character. The wave function of the electron ψ(r,t ) is obtained by solving Schrodinger’s wave equation. The probability of finding an electron is high where | ψ(r,t )|² is greater. Not only the information about the electron’s position but information about all the properties including energy etc. that we calculated using the Bohr’s postulates are contained in the wave function of ψ(r,t).
Quantum Mechanics of the Hydrogen Atom
The wave function of the electron ψ(r,t) is obtained from the Schrodinger’s equation
-(h²/8π²m) [∂²ψ /∂x² + ∂²ψ /∂y² + ∂²ψ/∂z²] - Ze²ψ/4πε0r = E ψ
where
(x.y,z ) refers to a point with the nucleus as the origin and r is the distance of this point from the nucleus.
E refers to the energy.
Z is the number of protons.
There are infinite number of functions ψ(r,t) which satisfy the equations.
These functions may be characterized by three parameters n,l, and ml.
For each combination of n,l, and ml there is an associated unique value of E of the atom of the ion.
The energy of the wave function of characterized by n,l, and ml depends only on n and may be written as
En = - mZ²e4/8 ε0²h²n²
These energies are identical with Bohr’s model energies.
The paramer n is called the principal quantum number, l the orbital angular momentum quantum number and ml. The magnetic quantum number.
When n = 1, the wave function of the hydrogen atom is
ψ(r) = ψ100 = √(Z³/ π a0²) *(e-r/ a0)
ψ100 denotes that n =1, l = 0 and ml = 0
a0 = Bohr radius
In quantum mechanics, the idea of orbit is invalid. At any instant the wve function is spread over large distances in space, and wherever ψ≠ 0, the presence of electron may be felt.
The probability of finding the electron in a small volume dV is | ψ(r)| ² dV
We can calculate the probability p(r)dr of finding the electron at a distance between r and r+dr from the nucleus.
In the ground state for hydrogen atom it comes out to be
P(r) = (4/ a0)r²e -2r/ a0
The plot of P(r) versus r shows that P(r) is maximum at r = a0 Which the Bohr’s radius.
But when we put n =2, the maximum probability comes at two radii one near r = a0 and the other at r = 5.4 a0. According to Bohr model all electrons should be at r = 4 a0.
Electron has a wave character as well as a particle character. The wave function of the electron ψ(r,t ) is obtained by solving Schrodinger’s wave equation. The probability of finding an electron is high where | ψ(r,t )|² is greater. Not only the information about the electron’s position but information about all the properties including energy etc. that we calculated using the Bohr’s postulates are contained in the wave function of ψ(r,t).
Quantum Mechanics of the Hydrogen Atom
The wave function of the electron ψ(r,t) is obtained from the Schrodinger’s equation
-(h²/8π²m) [∂²ψ /∂x² + ∂²ψ /∂y² + ∂²ψ/∂z²] - Ze²ψ/4πε0r = E ψ
where
(x.y,z ) refers to a point with the nucleus as the origin and r is the distance of this point from the nucleus.
E refers to the energy.
Z is the number of protons.
There are infinite number of functions ψ(r,t) which satisfy the equations.
These functions may be characterized by three parameters n,l, and ml.
For each combination of n,l, and ml there is an associated unique value of E of the atom of the ion.
The energy of the wave function of characterized by n,l, and ml depends only on n and may be written as
En = - mZ²e4/8 ε0²h²n²
These energies are identical with Bohr’s model energies.
The paramer n is called the principal quantum number, l the orbital angular momentum quantum number and ml. The magnetic quantum number.
When n = 1, the wave function of the hydrogen atom is
ψ(r) = ψ100 = √(Z³/ π a0²) *(e-r/ a0)
ψ100 denotes that n =1, l = 0 and ml = 0
a0 = Bohr radius
In quantum mechanics, the idea of orbit is invalid. At any instant the wve function is spread over large distances in space, and wherever ψ≠ 0, the presence of electron may be felt.
The probability of finding the electron in a small volume dV is | ψ(r)| ² dV
We can calculate the probability p(r)dr of finding the electron at a distance between r and r+dr from the nucleus.
In the ground state for hydrogen atom it comes out to be
P(r) = (4/ a0)r²e -2r/ a0
The plot of P(r) versus r shows that P(r) is maximum at r = a0 Which the Bohr’s radius.
But when we put n =2, the maximum probability comes at two radii one near r = a0 and the other at r = 5.4 a0. According to Bohr model all electrons should be at r = 4 a0.
Thursday, January 1, 2009
Ask questions and answer questions about IIT JEE Subjects
KNOWLEDGE QUESTION AND ANSWER BOARD
http://knol.google.com/k/narayana-rao-kvss/-/2utb2lsm2k7a/654#
http://knol.google.com/k/narayana-rao-kvss/-/2utb2lsm2k7a/654#
Sunday, December 7, 2008
Transmutation of elements
Question
Was it found by Rutherford and Chadwick that oxygen and carbon cannot be transmuted by bombarding them with alpha particles???
I searched internet and got this one.
Russell was able to demonstrate the transmutation of gases in the Bloomfield, New Jersey research laboratory at the Westinghouse Lamp Company on Sept. 30, 1927. Transmutation of hydrogen and oxygen to nitro- gen, and, nitrogen to oxygen and hydrogen was accomplished.
See reference
http://padrak.com/ine/NEN_4_11_3.html
Was it found by Rutherford and Chadwick that oxygen and carbon cannot be transmuted by bombarding them with alpha particles???
I searched internet and got this one.
Russell was able to demonstrate the transmutation of gases in the Bloomfield, New Jersey research laboratory at the Westinghouse Lamp Company on Sept. 30, 1927. Transmutation of hydrogen and oxygen to nitro- gen, and, nitrogen to oxygen and hydrogen was accomplished.
See reference
http://padrak.com/ine/NEN_4_11_3.html
Monday, November 10, 2008
Early Atomic Models
The idea that all matter is made of very small indivisible particles is very old.
Robert Boyle’s study of compression and expansion of air brings out the idea that air is made of tiny particles with lot of empty space between the particles.
The smallest unit of an element which carries all the properties of the element is called an atom.
Experiments on discharge tube, measurement of e/m by Thomson etc. established the existence of negatively charged electrons in the atoms.
Because atoms are electrically neutral, a search for the positive charge inside the atom was started.
Robert Boyle’s study of compression and expansion of air brings out the idea that air is made of tiny particles with lot of empty space between the particles.
The smallest unit of an element which carries all the properties of the element is called an atom.
Experiments on discharge tube, measurement of e/m by Thomson etc. established the existence of negatively charged electrons in the atoms.
Because atoms are electrically neutral, a search for the positive charge inside the atom was started.
Thomson’s Model of the atom
Thomson (1898) suggested that the atom is a positively charged solid sphere in which electrons are embedded in sufficient number to create a neutral atom. This model of the atom could explain why only negatively charged particles are being emitted when a metal is heated. This model was also useful to explain the formation of ions and ionic compounds.
Lenard's Suggestion on Atomic Structure
Lenard observed that cathode rays are passing through thin material without any deviation. According to him, this was so because, there is a lot of empty space in atoms. Hence the positive charged particles are also tiny like electrons.
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