chapter4

chapter4 - Chapter 4 Schroedinger Equation Einsteins...

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Chapter 4 Schroedinger Equation Einstein’s relation between particle energy and frequency Eq.(3.83) and de Broglie’s relation between particle momentum and wave number of a corre- sponding matter Eq.(3.84) suggest a equation for matter waves. This search for an equation describing matter waves was carried out by Erwin Schroedinger. He successful in the year 1926. The energy of a classical, nonrelativistic particle with momentum that is subject to a conservative force derived from a potential V ( ± r ) is 2 E = + V ( ± r ) . (4.1) 2 m For simplicity lets begin rst with a constant potential V ( ± r )= V 0 = const. This is the force free case. According to Einstein and de Broglie, the dis- persion relation between ω and ± k for describing the particle motion should be ~ 2 ± k 2 ~ ω = + V 0 . (4.2) 2 m Note, so far we had a dispersion relation for in one dimension, where the wavenumber k ( ω ) , a function of frequency. in three dimen- sions the frequency of the is rather a function of the three components o f thewavevector . Eachwavew ithag ivenwavevector ± k has the following time dependence e j ( ± k · ± r ωt ) , with ω = ~ ± k 2 + V 0 (4.3) 2 m ~ Note, thisisawave with phasefrontstravelingto the right. In contrast to our notation used in chapter 2 for rightward traveling electromagnetic waves, 199
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200 CHAPTER 4. SCHROEDINGER EQUATION sw itched thes igninthe exponent . Th isnotat ioncon formsw iththe phys ics oriented literature. A superposition of such waves in ± k space enables us to construct wave packets in real space Z ³ ´ Ψ ( ± r, t )= φ ω ± k,ω e j ( ± k · ± r ωt ) d 3 kdω (4.4) The inverse transform of the above expression is ³ ´ Z ± φ ω ± k, ω = 1 4 Ψ ( ± t ) e j ( ± k · r ωt ) d 3 rd t , (4.5) (2 π ) with à ! ³ ´ ~ ± k 2 V 0 φ ω ± ω = φ ( k ) δ ω 2 m ~ . (4.6) Or we can rewrite the function in Eq.(4.4) by carrying out the trivial frequency integration over ω à " à ! #! Z Ψ ( ± t φ ( k )exp j ± k · ± r ~ 2 ± k m 2 + V ~ 0 t d 3 k. (4.7) Due to the Fourier relationship between the function in space and time coordinates and the function in vector and frequency coordinates ³ ´ φ ω ± ω Ψ ( ± t ) (4.8) have ωφ ω ( ω ) j Ψ ( ± t ) , (4.9) ∂t ± k φ ω ( ω ) ↔− j Ψ ( ± t ) , (4.10) ± k 2 φ ω ( ω ) ∆Ψ ( ± t ) . (4.11) where = ±e x ∂x + y ∂y + z ∂z , (4.12) 2 2 2 = · ∇≡∇ 2 = 2 + 2 + 2 . (4.13)
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201 From the dispersion relation follows by multiplication with the wave function in thewavevectorand frequencydoma ~ 2 k 2 ~ ωφ ω ( k, ω )= φ ω ( ω )+ V 0 φ ω ( ω ) . (4.14) 2 m With the inverse transformation the corresponding equation in the space and tieme domain is j ~ Ψ ( ± r, t ) = ~ 2 ∆Ψ ( ± t V 0 Ψ ( ± t ) . (4.15) ∂t 2 m Generalization of the above equation for a constant potential to the instance of an arbitrary potential in space leads nally to the Schroedinger equation j ~ Ψ ( ± t ) = ~ 2 ( ± t V ( ± r ) Ψ ( ± t ) . (4.16) 2 m Note, the last few pages ar not a derivation of the Schroedinger Equation but rather a motivation for it based on the ndings of Einstein and deBroglie.
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This note was uploaded on 12/02/2010 for the course ECE 6.641 taught by Professor Zahn during the Spring '09 term at MIT.

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chapter4 - Chapter 4 Schroedinger Equation Einsteins...

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