1000 Solved Problems in Modern Physics

124 2 Quantum Mechanics – I 2.68 (a) Letf=eiA; thenf†= ( eiA )† =e−iA Thereforef†f=e−iAeiA= 1 ThuseiAis unitary. (b) (a) Momentu ...

2.3 Solutions 125 Integration of (1) yields ∫ ∂ ∂x (ψ∗ψ)dτ+ ∫ ψ∗ ∂ψ ∂x dτ+ ∫ ψ dψ∗ dx dτ (2) The integral on the LHS vanishes be ...

126 2 Quantum Mechanics – I This givesa=1,b=0forλ=1 anda=0,b=1forλ= 3 Hence the eigen states ofAare ( 1 0 ) and ( 0 1 ) (c) AsA= ...

2.3 Solutions 127 The eigen vector associated withλ 1 =1is |ψ 1 >= ∑^2 n= 1 Cn|n>with−C 1 −iC 2 = 0 ,C 2 =iC 1 , |ψ 1 > ...

128 2 Quantum Mechanics – I 2.3.8 Uncertainty Principle............................... 2.82ΔxΔpx∼/ 2 P=  2 x E= p^2 2 m + 1 / ...

2.3 Solutions 129 The precise statement of the Heisenberg uncertainty principle is ΔPxΔx≥/ 2 ΔPyΔy≥/2(3) ΔPzΔz≥/ 2 Consider t ...

130 2 Quantum Mechanics – I This value is in agreement with 13.60 obtained from Bohr’s theory of hydro- gen atom. 2.86 (i)ΔxΔPx= ...

Chapter 3 Quantum Mechanics – II 3.1 Basic Concepts and Formulae ................................ Schrodinger’s equation i ∂ψ ∂ ...

132 3 Quantum Mechanics – II Table 3.1Dynamic quantities and operators Physical Quantity Operator Position r R Momentum P −i∇ K ...

3.1 Basic Concepts and Formulae 133 Commutators AB−BA=[A,B] (3.12) by definition. Dirac’s Bra and Ket notation A ket vector, or ...

134 3 Quantum Mechanics – II Table 3.2Some selected eigen functions of hydrogen atom State NLm u 1S 1 0 0 Ane−x 2S 2 0 0 Ane−x(1 ...

3.1 Basic Concepts and Formulae 135 Hetero-nuclear diatomic molecules such as CO and linear polyatomic molecule such as HCN, do ...

136 3 Quantum Mechanics – II with a trial functionψthat depends on a number of parameters, and varying these parmeters until the ...

3.2 Problems 137 The Born approximation Here the entire potential energy of interaction between the colliding particles is regar ...

138 3 Quantum Mechanics – II 3.8 (a) State and explain Laporte rule for light emission. (b) What are metastable states? 3.9 Show ...

3.2 Problems 139 (a) PutF(r)=exp (−r/ν)y(r), whereE=− 1 /(2ν^2 ), and show that d^2 y dx^2 = 2 ν d dr − ν r y (b) Assuming thaty ...

140 3 Quantum Mechanics – II 3.17 In Problem 3.16, (a) Consider the case wherem=0. Make the change of variableμcosθand consider ...

3.2 Problems 141 3.22 The small binding energy of the deuteron (2.2 MeV) implies that the maximum ofU(r) lies just inside the ra ...

142 3 Quantum Mechanics – II Use the constants to represent the amplitude of the reflected and trans- mitted particle streams re ...

3.2 Problems 143 Fig. 3.2Bound states in a square well potential 3.36 In Problem 3.25 express the normalization constantAin term ...