1000 Solved Problems in Modern Physics

84 1 Mathematical Physics N=6; ∑ yn= 29 .5; ∑ xn=3; ∑ x^2 n= 19 ∑ xn^3 =27; ∑ xn^4 =115; ∑ xnyn= 21 .3; ∑ xn^2 yn= 158. 1 Solvin ...

1.3 Solutions 85 1.103 <x>= ∫∞ 0 xf(x)dx/ ∫∞ 0 f(x)dx = ∫∞ 0 x^2 e− xλ dx/ ∫∞ 0 xe− xλ dx = 2 λ^3 λ^2 = 2 λ Most probable ...

86 1 Mathematical Physics The ordinates are found by substitutingx = 0 , 1 , 2 ···10 in the equation y=x^2. Thus area= 1 3 (0+ 4 ...

Chapter 2 Quantum Mechanics – I 2.1 Basic Concepts and Formulae Wave number ν ̃= 1 λ (2.1) 1fm= 10 −^15 m; 1A ̊ = 10 −^10 m; 1 n ...

88 2 Quantum Mechanics – I Natural units =c= 1 (2.7) Mosley’s law (for characteristic X-rays) √ ν=A(Z−b) (2.8) whereZis the ato ...

2.1 Basic Concepts and Formulae 89 whereμis the reduced mass given by μ= memp me+mp (2.17) andZ=1 for H-atom. The radius of the ...

90 2 Quantum Mechanics – I Table 2.1Hydrogen spectrum Series Region First line (A) ̊ Series limit (A) ̊ Lyman Ultraviolet 1,215 ...

2.1 Basic Concepts and Formulae 91 parallel to the light path), they are split into two, which are circularly polarized in the o ...

92 2 Quantum Mechanics – I 2.2 Problems.................................................. 2.2.1 deBroglieWaves 2.1 (a) Write dow ...

2.2 Problems 93 What additional quantum numbers are needed to specify fully an atomic quan- tum state and what physical quantiti ...

94 2 Quantum Mechanics – I 2.14 Calculate the wavelengths of the first four lines of the Lyman series of the positronium on the ...

2.2 Problems 95 2.21 A spectral line of atomic hydrogen has its wave number equal to the difference between the two lines of Bal ...

96 2 Quantum Mechanics – I 2.33 Moseley’s law for characteristic x-rays is of the form √ v=a(z−b). Calculate the value of a forK ...

2.2 Problems 97 ground state and their mean kinetic energy was 2kT,estimate the magnetic moment. [Adapted from the University of ...

98 2 Quantum Mechanics – I 2.47 The^9 Be+ion has a nucleus with spinI = 3 /2. What values are possible for the hyperfine quantum ...

2.2 Problems 99 2.56 In the Helium-Neon laser (three-level laser), the energy spacing between the upper and lower levelsE 2 −E 1 ...

100 2 Quantum Mechanics – I 2.65 TheJ= 0 →J=1 rotational absorption line occurs at wavelength 0.0026 in C^12 O^16 and at 0.00272 ...

2.3 Solutions 101 2.78 Show that (σ.A)(σ.B)=A.B+iσ.(A×B) whereAandBare vectors andσ’s are Pauli matrices. 2.79 The Pauli matrixσ ...

102 2 Quantum Mechanics – I cPγ=Eγ= 67 .5MeV λ= h p = 2 π c cp = (2π)(197.3MeV.fm) 67 .5MeV = 18 .36 fm= 1. 836 × 10 −^14 m. 2. ...

2.3 Solutions 103 νp= ω k = ( c^2 k^2 + m^2 c^4 ^2 ) 1 / 2 /k νg= dω dk =kc^2 ( c^2 k^2 + m^2 c^4 ^2 )− 1 / 2 ∴νpνg=c^2 2.3.2 ...