Modern Physics

 Expectation value; Expectation value. Expectation value.

 Operators: Operator is a rule or set of rules which when applied to a function within a given domain produces another function within a given range. For example, when differential operator is applied to a function f(x) it changes into another function g(x) within a given range. In quantum mechanics, since the momentum and energy cannot be measured accurately due to uncertainty principle. The operator are important to find the average or expectation values of energy, momentum etc. The various operators are given below. 1) Energy operator. 2) Momentum operator. 3) kinetic energy operator. 4) Hamiltonian operator. 5) Angular momentum operator. Different operators.  Different operators.

 Linearity and Superposition: Wave functions add, not probabilities. The important property of Schrodinger's equation is that it is linear in wavefunction Ψ. By this we meant that the equation is in terms of Ψ and its derivatives, and no term is independent of Ψ or involves higher powers of Ψ or its derivatives . As a result, the linear combination of solutions of Schrodinger's equation is itself a solution. If Ψ₁ and Ψ₂ are solutions of Schrodinger's equation then Ψ=a₁Ψ₁+a₂Ψ₂ is itself a solution. Where a₁ and a₂ are constants. Thus wavefunction Ψ₁ and Ψ₂ obeys superposition principle. Let us apply superposition principle to the diffraction of electron beam. The fig(a) represents two slits (1&2) from which a beam of electrons passes to a viewing screen. If the slit 1 is open as shown in fig (b), then the corresponding probability density is P₁=|Ψ₁|² = Ψ₁*Ψ₁ If the slit 2 is open as shown in fig (c), then the corresponding probability density is P₂ = |Ψ₂|² = Ψ₂*Ψ₂ We

 Schrodinger's wave equation (Time Independent) Time Independent Schrodinger's wave equation. Time Independent Schrodinger's wave equation.

Applications of the Heisenberg's uncertainty principle; The applications of Heisenberg 's uncertainty principle are given below; 1) Non-Existence of electron in the nucleus. We know that the radius of the nucleus of any atom is of the order of 10⁻¹⁴m, so if an electron is confined within the nucleus , the uncertainty in its position must not be greater than 10⁻¹⁴m. According to uncertainty principle , ∆x.∆p ≈ ℏ/2 Where ∆x = uncertainty in the position. ∆p = uncertainty in the momentum. And   ℏ = h/2π Since ∆x=r=10⁻¹⁴ ∴ ∆p≈h/4π∆x = 6.62×10⁻³⁴/4×3.14× 10⁻¹⁴         =5.27×10⁻²¹ kgms⁻¹ If this is the uncertainty in the momentum of the electron, the momentum of the electron must be at least comparable with its magnitude, i.e p ≈ 5.27×10⁻²¹ kgms⁻¹ The kinetic energy of the electron of mass m(=9.1×10⁻³¹kg) is given by E = p²/2m ≈ [5.27× 10⁻²¹]²/ 2×9.1×10⁻³¹J E = [5.27×10⁻²¹]²/2×9.1×10⁻³¹×1.6×10⁻¹³ MeV    ≈ 95.5 MeV [∴1MeV=1.6×10⁻¹³J] It means that if the electron exist inside the nucl