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 wave Eq.(3.84) suggest a wave equation for matter waves. This search for an equation describing matter waves was carried out by Erwin Schroedinger. He was successful in the year 1926. The energy of a classical, nonrelativistic particle with momentum p that is subject to a conservative force derived from a potential V ( r ) is 2 E = p + V ( r ) . (4.1) 2 m For simplicity lets begin fi 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 waves 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 waves in one dimension, where the wavenumber k ( ω ) , was a function of frequency. For waves in three dimen - sions the frequency of the wave is rather a function of the three components of the wave vector. Each wave with a given wave vector k has the following time dependence e j ( k · r ωt ) , with ω = ~ k 2 + V 0 (4.3) 2 m ~ Note, this is a wave with phase fronts traveling to the right. In contrast to our notation used in chapter 2 for rightward traveling electromagnetic waves, we 199
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200 CHAPTER 4. SCHROEDINGER EQUATION switched the sign in the exponent. This notation conforms with the physics 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 k dω (4.4) The inverse transform of the above expression is ³ ´ Z φ ω k, ω = 1 4 Ψ ( r, t ) e j ( k · r ωt ) d 3 r dt, (4.5) (2 π ) with à ! ³ ´ ~ k 2 V 0 φ ω k, ω = φ ( k ) δ ω 2 m ~ . (4.6) Or we can rewrite the wave function in Eq.(4.4) by carrying out the trivial frequency integration over ω à " à ! #! Z Ψ ( r, 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 wave function in space and time coordinates and the wave function in wave vector and frequency coordinates ³ ´ φ ω k, ω Ψ ( r, t ) (4.8) we have ω φ ω ( k, ω ) j Ψ ( r, t ) , (4.9) ∂t k φ ω ( k, ω ) j Ψ ( r, t ) , (4.10) k 2 φ ω ( k, ω ) ∆ Ψ ( r, t ) . (4.11) where = e x ∂x + e y ∂y + e z ∂z , (4.12) 2 2 2 = · ∇ ≡ ∇ 2 = ∂x 2 + ∂y 2 + ∂z 2 . (4.13)
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201 From the dispersion relation follows by multiplication with the wave function in the wave vector and frequency domain ~ 2 k 2 ~ ω φ ω ( k, ω ) = φ ω ( k, ω ) + V 0 φ ω ( k, 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 ( r, t ) + V 0 Ψ ( r, 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 fi nally to the Schroedinger equation j ~ Ψ ( r, t ) = ~ 2 ∆ Ψ ( r, t ) + V ( r ) Ψ ( r, t ) . (4.16) ∂t 2 m Note, the last few pages ar not a derivation of the Schroedinger Equation but rather a motivation for it based on the fi ndings of Einstein and deBroglie.
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