Modern Physics. Ans: 1. According to de broglie hypothesis, any moving particle is associated with a wave.

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Dominic Stewart
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1 Modern Physics Q1. What is matter wave? Or What is De broglie hypothesis? Ans: 1. According to de broglie hypothesis, any moving particle is associated with a wave. 2. The waves associated with a particle is called matter wave/de broglie wave. 3.Wavelength associated with the matter wave is called de broglie wavelength. h λ= Or λ = h mv p m= mass of the particle v= velocity of the particle p=mv= momentum of the particle. Q2. Derive the expression for de-broglie wavelgth. Ans: E = hν=hc/ hv/ E = mc2 = mv =2 hv/ = mv2. Cancel v. h/ = mv since h/ = p = h/p mv = p.

2 Q3. Expression for de broglie wavelgth in different forms.

waves. Ans: 1. Experimental arrangement consist of filament, Ni crystal, detector. 2.

3 Q4. Write the properties of Matter waves. Q5: Describe the Davission Germer expriment for the proof of wave nature of matter Or Describe the Davission Germer expriment for the exprimental evidence of matter waves. Ans: 1. Experimental arrangement consist of filament, Ni crystal, detector. 2. Electron beam (electrons) are generated from the filament by passing high current through it. When filament gets heated up, it emits electrons. 3. Emitted electrons are accelerated by potential (voltage) applied between filament and the anode. 4. Accelerated electron beam is allowed to fall on the Ni crystal. 5. These electrons are scattered by Ni crystal in all the directions.

6. The detector D measures the number of electrons scattered by the crystal in different directions/angles (0 90 degrees). 7. Experiment was performed for different accelerated potentials. 8.

4 6. The detector D measures the number of electrons scattered by the crystal in different directions/angles (0 90 degrees). 7. Experiment was performed for different accelerated potentials. 8. At 54V accelearting voltage, intensity graph have shown below. It is cleared that at 50 degrees, a intense hump (intense peak) is observed. This indicates that electrons are scattered more. This behaviour is not observed for other accelerating voltages. Intensity Peak /hump At 54 V Ф Ф=50 Ф O 9. If you calculate wave length associated with electrons at 54 V = 1.66 Å From below figure, we can measure angle of incidence of electron beam on the Ni crystal planes and it is found that 65 degrees. Or 11. Substitute above values in braggs law of x-ray diffraction According to braggs law 2dsinθ=nλ d=0.91 Å, n=1, θ=65o λ= 1.65 Å from the values of wavelength obtianed by equation 1 (De-broglie) and 2 (braggs) are same/agreed well.

$13. Therefore, this experiment gave a evidence that electrons exhibit diffraction (wave nature). This implies the existance of matter waves.$ Experimental arrangment consist of Filament (F), Anode (A), Photograhic plate (P), Thin gold foil (G) (Poly crystalline) and Metal block (B). 2.

Experimental arrangment consist of Filament (F), Anode (A), Photograhic plate (P), Thin gold foil (G) (Poly crystalline) and Metal block (B). 2.

Electron beam passes through a fine hole in metal block (B) and falls on the gold foil of thickness 0.1 μm. The electron are passing through foils are received on the photographic plate (P).

5 13. Therefore, this experiment gave a evidence that electrons exhibit diffraction (wave nature). This implies the existance of matter waves. Q6: Describe the G P Thomson's expriment for the proof of wave nature of matter Or Describe the G P Thomson's expriment for the exprimental evidence of matter waves. Ans: 1. Experimental arrangment consist of Filament (F), Anode (A), Photograhic plate (P), Thin gold foil (G) (Poly crystalline) and Metal block (B). 2. Electrons are produced from the heated filament (F) and accelerated through high potentail given to the anode (A). 3. Electron beam passes through a fine hole in metal block (B) and falls on the gold foil of thickness 0.1 μm. The electron are passing through foils are received on the photographic plate (P). 4. Metals are poly crystalline in which grains are oriented randomaly and some grains always satisfies braggs law with respect to the incident electron beam angle and produce the braggs reflection. 5. A concentric ring pattern is produced on the photographic plate like as shown in below. 6. X-ray diffraction pattern of powder (poly crystalline) samples is shown in below.

$Form above two figures, the diffraction pattern produced by electron beam is similar to the x-ray diffraction pattern produced using x-rays.$

6 Form above two figures, the diffraction pattern produced by electron beam is similar to the x-ray diffraction pattern produced using x-rays. Therefore, this experiment provides the evidence for the wave nature of electron. Which implies the existance/evidence of matter waves. Q7. Explain the terms (a) Wave packet (b) Group velocity (c) Phase velocity. Ans: (a) Wavepacket: It is representation of matter wave associated with a particle. Which is represnted bleow Wave packet is the resultant of superpostion of large number of harmonic waves slightly differ in frequency (b) Group velocity (Vg): The velocity with which wave packet advances (moving) in the medium is called group velocity. Vg = dω dk

7 (c) Phase velocity (Vp): The individual waves forming the wave packet propogate at a velocity known as the phase velocity (V p) Vp= ω/k Q8. What is Heisenberge uncertainity applications principle and write its Ans: Principle: It is impossible to measure both the position and momentum of a particle simulataneously and precisely. Δp Δx Other forms Application 1 h 4π ΔE Δt h 4π

8 Application 2 Application 3 Particle in a box : Let us consider a particle confined to a box of length l. The uncertainity Δx in the position is l Δx.ΔP h ΔP=h/l ( h cut/h bar) ( Δx=l )

9 E= P2/2m E= (ΔP)2/2m E= h2/2ml This Energy result agrees with the result obtained from the schrodinger wave equation. Q9. Derive the Schrodinger s equation. Ans: time independent wave

12 Q 10 Write physical significance of wave function. Ans: 1. It gives a statistical relationship between the particle and wave. 2. The probability of finding a particle within a volume dv P= ΨΨ* dv= Ψ 2dv dv=dxdydz, Ψ2 Gives the probability finding a particle. 3. When the particle definately exist in a volume P= 4. ΨΨ* dv=1 (Normalization condition ) When the particle does not exist in a volume P= ΨΨ* dv=0

13 Q11. Derive Schrodinger Equation for a particle in a one dimensional "box" and find its energy (eigen ) values, wave (eigen) functions and probability.

17 Q.12 Explain the formation of energy bands in solids

19 Q.13 Distingush between insulator, conductor, and semiconductor. 1. Conductors

20 2. Semiconductors 3. Insulator

21 Q. 14 Explain in detail the comparison among Maxwell Boltzmen (M-B), Fermi Dirac (F-D) and Bose-Einstein (BE) statistics

Q. 15 Define the tunneling phenomenon and find the transmission and reflection coefficient of the partcile (electron) in case of rectangular potential barrier.

22 Q. 15 Define the tunneling phenomenon and find the transmission and reflection coefficient of the partcile (electron) in case of rectangular potential barrier. Or Explain the tunneling phenomenon in case of rectangular potential barrier. Ans: Defination : Quantum tunnelling or tunneling refers to the quantum mechanical phenomenon where a particle tunnels through a barrier that it classically could not not possible like as shown in the figure. Rectangular potential barrier: V=0 V=0 V(x)=V0 Region - I Region -III Region -II X=0 X=L

24 The-end

Quantum Mechanics Tutorial

Quantum Mechanics Tutorial The Wave Nature of Matter Wave-particle duality and de Broglie s hypothesis. de Broglie matter waves The Davisson-Germer experiment Matter wave packets Heisenberg uncertainty