Introduction to Elementary Particles

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Lecture 5 30 Oct 23, 2002 4.2.2 Examples 1. One-dimensional infinite square well ; see the figure. In practice, this situation is hardly realizable: infinitely steep “walls” simply do not occur in nature. However, it is a nice approximation for many real-life occurrences. The infinite square well potential is formally given by: (I.82) 0 for 0 () everywhere else xL Ux ≤≤ = L 0 x U ( x ) The Schrödinger equation will now take on a different form inside and outside the infinite well: 22 0 2/ 2 0 2 0: 0 ( ) ( ) s i nc o s 2 0, : ( ) ( ) 0 2() km E x xLU x x A k xB k x mE x x xx L U x x mx ϕ ϕϕ = = =− = + <> = = ⇒ = ∞∂ = = = (I.83) Thus, similar to a classical particle, a quantum particle cannot exist outside the well. Inside the well, the particle wave function solution contains two constants A and B ; the wave number k is given by the a priori unknown energy E of the particle. The challenge is to determine the wave function constants ánd the value E in the problem. We have the following constraints: on both edges of the well, the wave function solutions left and right of the potential step have to smoothly match: 2 0 ( 0) ( 0) and ( ) ( ) ( 0 and ( ) 0 1 L x L L xd x ↑= ⇒= = = = = x L (I.84) The first two so-called “boundary conditions” give us two equations. Moreover, we get an addi- tional normalization constraint (the third equation) from the fact that there is one particle in the well, i.e. the probability for finding the particle anywhere in the well is 1. These three equations suffice to give A , B , and E : 2 2 non-relativistic 2 2 00 (0) 0 ( ) sin 0 with 1,2,. .. ; ;, o r i n t e r m s o f : 8 2 Normalization: 1 ( ) sin 2 nn n n LL B L A kL kL n n Lp h h kn E E n Ln m m m L nx L x A d x A A L L ϕπ ππ λ λλ π == = = = =  = =   ∫∫ (I.85) So, apart from an unknown phase factor, the wave function ánd a discrete energy spectrum are found from Schrödinger equation! The allowed energies are a discrete set of energy levels , quadratically increasing (in the non-relativistic limit) with a natural number n , the quantum number in this problem. For ultra-relativistic particles: E n = p n c = hc / n = nhc /(2 L ). The solutions (i.e. the wave functions) are simple standing waves with zero, one, two, etc. nodes inside the well, and have value zero at the edges. For the ground state 1 ( x ) = (2/ L )
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Lecture 5 31 Oct 23, 2002 sin( k 1 x ) = (2/ L ) sin( x π / L ). The probability for finding the particle as function of x (inside the well) equals P ( x ) = (2/ L ) sin 2 ( x / L ), i.e. it is maximum near the center.
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Lecture 05 - Lecture 5 30 Oct 23, 2002 4.2.2 1....

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