Matter waves (de-Broglie Waves).
According to de-Broglie a moving material particle sometimes acts as a wave and sometimes as a particle.
A wave is associated with moving material particle which control the particle in every respect.
The wave associated with moving particle is called matter wave or de-Broglie wave and it propagates in the form of wave packets with group velocity.
(1) de-Broglie wavelength
According to de-Broglie theory, the wavelength of de-Broglie wave is given by
Where h = Plank’s constant, m = Mass of the particle, v = Speed of the particle, E = Energy of the particle.
The smallest wavelength whose measurement is possible is that of -rays.
The wavelength of matter waves associated with the microscopic particles like electron, proton, neutron, a-particle etc. is of the order of m.
(i) de-Broglie wavelength associated with the charged particles.
The energy of a charged particle accelerated through potential difference V is
Hence de-Broglie wavelength
Å, Å, Å, Å
(ii) de-Broglie wavelength associated with uncharged particles.
For Neutron de-Broglie wavelength is given as Å
Energy of thermal neutrons at ordinary temperature
; where k = Boltzman’s constant = Joules/kelvin , T = Absolute temp.
(2) Some graphs
Note : q A photon is not a material particle. It is a quanta of energy.
When a particle exhibits wave nature, it is associated with a wave packet, rather then a wave.
(3) Characteristics of matter waves
(i) Matter wave represents the probability of finding a particle in space.
(ii) Matter waves are not electromagnetic in nature.
(iii) de-Brogile or matter wave is independent of the charge on the material particle. It means, matter wave of de-Broglie wave is associated with every moving particle (whether charged or uncharged).
(iv) Practical observation of matter waves is possible only when the de-Broglie wavelength is of the order of the size of the particles is nature.
(v) Electron microscope works on the basis of de-Broglie waves.
(vi) The electric charge has no effect on the matter waves or their wavelength.
(vii) The phase velocity of the matter waves can be greater than the speed of the light.
(viii) Matter waves can propagate in vacuum, hence they are not mechanical waves.
(ix) The number of de-Broglie waves associated with nth orbital electron is n.
(x) Only those circular orbits around the nucleus are stable whose circumference is integral multiple of de-Broglie wavelength associated with the orbital electron.
(4) Davision and Germer experiment
It is used to study the scattering of electron from a solid or to verify the wave nature of electron. A beam of electrons emitted by electron gun is made to fall on nickel crystal cut along cubical axis at a particular angle. Ni crystal behaves like a three dimensional diffraction grating and it diffracts the electron beam obtained from electron gun.
The diffracted beam of electrons is received by the detector which can be positioned at any angle by rotating it about the point of incidence. The energy of the incident beam of electrons can also be varied by changing the applied voltage to the electron gun.
According to classical physics, the intensity of scattered beam of electrons at all scattering angle will be same but Davisson and Germer, found that the intensity of scattered beam of electrons was not the same but different at different angles of scattering.
Intensity is maximum at 54 V potential difference and 50o diffraction angle.
If the de-Broglie waves exist for electrons then these should be diffracted as X-rays. Using the Bragg’s formula , we can determine the wavelength of these waves.
Where d = distance between diffracting planes, = glancing angle for incident beam = Bragg’s angle.
The distance between diffraction planes in Ni-crystal for this experiment is d = 0.91Å and the Bragg’s angle = 65o. This gives for n = 1, Å
Now the de-Broglie wavelength can also be determined by using the formula .
Thus the de-Broglie hypothesis is verified.
Heisenberg Uncertainty Principle.
According to Heisenberg’s uncertainty principle, it is impossible to measure simultaneously both the position and the momentum of the particle.
Let Dx and Dp be the uncertainty in the simultaneous measurement of the position and momentum of the particle, then ; where and h = 6.63 ´ 10–34
J-s is the Planck’s constant.
If Dx = 0 then Dp = ¥
and if Dp = 0 then Dx = ¥
i.e., if we are able to measure the exact position of the particle (say an electron) then the uncertainty in the measurement of the linear momentum of the particle is infinite. Similarly, if we are able to measure the exact linear momentum of the particle i.e., Dp = 0, then we can not measure the exact position of the particle at that time.
According to Eienstein’s quantum theory light propagates in the bundles (packets or quanta) of energy, each bundle being called a photon and possessing energy.
(1) Energy of photon
Energy of each photon is given by where c = Speed of light, h = Plank’s constant = 6.6 ´ 10–34
= Frequency in Hz, l = Wavelength of light
Energy of photon in electron volt
(2) Mass of photon
Actually rest mass of the photon is zero. But it’s effective mass is given as
. This mass is also known as kinetic mass of the photon
(3) Momentum of the photon
(4) Number of emitted photons
The number of photons emitted per second from a source of monochromatic radiation of wavelength l and power P is given as ; where E = energy of each photon
(5) Intensity of light (I)
Energy crossing per unit area normally per second is called intensity or energy flux
At a distance r from a point source of power P intensity is given by
It is the phenomenon of emission of electrons from the surface of metals, when light radiations (Electromagnetic radiations) of suitable frequency fall on them. The emitted electrons are called photoelectrons and the current so produced is called photoelectric current.
This effect is based on the principle of conservation of energy.
(1) Terms related to photoelectric effect
(i) Work function (or threshold energy) (W0) : The minimum energy of incident radiation, required to eject the electrons from metallic surface is defined as work function of that surface.
n0 = Threshold frequency; l0 = Threshold wavelength
Work function in electron volt W0(eV)
Note : q By coating the metal surface with a layer of barium oxide or strontium oxide it’s work function is lowered.
(ii) Threshold frequency (n0) : The minimum frequency of incident radiations required to eject the electron from metal surface is defined as threshold frequency.
If incident frequency n < n0
Þ No photoelectron emission
(iii) Threshold wavelength (l0) : The maximum wavelength of incident radiations required to eject the electrons from a metallic surface is defined as threshold wavelength.
If incident wavelength l > l0
Þ No photoelectron emission
(2) Einstein’s photoelectric equation
According to Einstein, photoelectric effect is the result of one to one inelastic collision between photon and electron in which photon is completely absorbed. So if an electron in a metal absorbs a photon of energy E (= hn), it uses the energy in three following ways.
(i) Some energy (say W) is used in shifting the electron from interior to the surface of the metal.
(ii) Some energy (say W0) is used in making the surface electron free from the metal.
(iii) Rest energy will appear as kinetic energy (K) of the emitted photoelectrons.
Hence E = W + W0 + K
For the electrons emitting from surface W = 0 so kinetic energy of emitted electron will be max.
Hence E = W0 + Kmax
; This is the Einstein’s photoelectric equation
(3) Experimental arrangement to observe photoelectric effect
When light radiations of suitable frequency (or suitable wavelength and suitable energy) falls on plate P, photoelectrons are emitted from P.
If plate Q is at zero potential w.r.t. P, very small current flows in the circuit because of some electrons of high kinetic energy are reaching to plate Q, but this current has no practical utility.
If plate Q is kept at positive potential w.r.t. P current starts flowing through the circuit because more electrons are able to reach upto plate Q.
As the positive potential of plate Q increases, current through the circuit increases but after some time constant current flows through the circuit even positive potential of plate Q is still increasing, because at this condition all the electrons emitted from plate P are already reached up to plate Q. This constant current is called saturation current.
To increase the photoelectric current further we will have to increase the intensity of incident light.
Photoelectric current (i) depends upon
Potential difference between electrodes (till saturation)
Intensity of incident light (I)
(c) Nature of surface of metal
To decrease the photoelectric current plate Q is maintained at negative potential w.r.t. P, as the anode Q is made more and more negative, fewer and fewer electrons will reach the cathode and the photoelectric current decreases.
At a particular negative potential of plate Q no electron will reach the plate Q and the current will become zero, this negative potential is called stopping potential denoted by V0.
If we increase further the energy of incident light, kinetic energy of photoelectrons increases and more negative potential should be applied to stop the electrons to reach upto plate Q. Hence .
Stopping potential depends only upon frequency or wavelength or energy of incident radiation. It doesn’t depend upon intensity of light.
We must remember that intensity of incident light radiation is inversely proportional to the square of distance between source of light and photosensitive plate P
i.e., so )
(4) Different graphs
(i) Graph between potential difference between the plates P and Q and photoelectric current
(ii) Graph between maximum kinetic energy / stopping potential of photoelectrons and frequency of incident light