chapter4

# chapter4 - Chapter 4 Schroedinger Equation Einsteins...

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 Ã ! 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 Ã " Ã ! #! 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)
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 41

chapter4 - Chapter 4 Schroedinger Equation Einsteins...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online