2 Lecture 2: The Bohr atom (1913) and the Schrödinger equation (1925)

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1 1 Lectue 1: The beginnings of quantum physics 1. The Sten-Gelach expeiment. Atomic clocks 3. Planck 1900, blackbody adiation, and E ω 4. Photoelectic effect 5. Electon diffaction though cystals, de Boglie 194, and p k 6. The Boh atom Lectue : The Boh atom 1913 and the Schödinge equation The Boh atom The Boh atom assumes the usual electostatic attaction between an electon and a poton, Then, fo an electon in a cicula obit, F ke ˆ a v ˆ To these classical elements, Boh added a quantization ule: the angula momentum must be a multiple of Planck s educed constant, L mv n Combining the classical elements, we have a elationship between the adius and velocity of cicula obits, ke mv Solving fo the velocity, we have Then accoding to the Boh quantization ule, v ke m n mv mke o, solving fo, n n mke The total enegy of the electon is 1

2 E 1 mv ke ke mk e 4 n 13.6eV n This means that the enegy of an electon that moves between two obits will change by E 1 mv ke ke mk e 4 n E n 1 m If this enegy is given off in the fom of a photon satisfying the Planck elation, then the fequency of the emitted light will be ω E A fomula of this fom had aleady been detemined expeimentally, and was now explained by the Boh model.. The Schödinge equation The Boh model esticts the electon to cicula motion in a plane, and gives incoect values of total angula momentum fo the electons. A fulle pictue was equied, and is povided by witing a 3-dimensional wave equation fo the electon. We may use the deboglie wavelength and the Planck elation, togethe with the elativistic elationship between enegy and momentum, to deive a suitable equation. We have: The 4-momentum of a paticle is given by E ω p k ev and the nom of this equation is p α mu α mγ c, v E c, p η αβ p α p β p α p α p 0 + p E c + p

3 On the othe hand, we have η αβ p α p β m u α u α m c Equating these, E c + p m c E p c + m c 4 Now suppose the electon is descibed by a plane wave, ψ Ae ik x ωt Then we may ecove the wave numbe and fequency by diffeentiation, ψ [ Ae ik x ωt] k ψ [ t ψ t Ae ik x ωt] ω ψ Multiplying each deivative by i, we have the enegy and momentum, Substituting these opeatos, into the enegy-momentum elation, i ψ k ψ p ψ i t ψ ω ψ p α Ec, p E ψ 1 i c t, i x α E p c + m c 4 i i c + m c 4 t and allowing this opeato elationship to act on a wave function, ψ, The diffeential opeato ψ t c ψ + m c 4 ψ 1 ψ c t + ψ m c ψ 1 c t + η αβ x α x β 3

4 is the spacetime genealization of the Laplacian, δ i x i x. The time dependence makes it a wave opeato, but because of the Planck and deboglie elationships, it also descibes paticle-like enegy and momentum. Indeed, the plane-wave solutions may be witten as The wave equation we have witten, ψ Ae i p x Et ψ m c ψ is called the Klein-Godon equation. It fist appeas in Schödinge s notes in 195 befoe being published the next yea fist by Oska Klein and Walte Godon, but also the same yea by Vladimi Fock, Johann Kuda, Théophile de Donde and Fans-H. van den Dungen, and Louis de Boglie. It is the obvious elativistic genealization of the Schödinge equation but fails to descibe electon spin. Additionally, because the equation is second ode in time deivatives, it equies both initial position and velocity specifications, and this is fobidden by the uncetainty pinciple. Finally, the equation leads to negative pobability states. In 195, Schödinge took a diffeent appoach. The poblems aising fom the second ode time deivatives may be avoided by fist solving fo the enegy, then taking a non-elativistic appoximation. We may then also add a potential to the enegy Fo v c we may expand so the non-elativistic vesion is 1 + p m c E p c + m c 4 + V mc 1 + p m c + V in a Taylo seies, 1 + p m c 1 + p m c p m c E mc 1 + p m c + V Making the same opeato substitutions that led us to the Klein-Godon equation, and allowing it to opeate on a function, φ, gives i φ t mc φ m φ + V φ The constant mass tem may be emoved by the eplacement φ ψe i mc t Then we find i ψe i mc t mc ψe i mct t m ψe i mc t + V ψe i mc t ψ i t e i mc t i mc ψe i mc t mc ψ m ψ + V ψ e i mc t esulting in the familia fom of the Schödinge equation, i ψ t m ψ + V ψ 4

5 .3 The Pauli equation Although this equation also fails to descibe the electon spin, Pauli genealized the Schödinge equation in 197. The esulting Pauli equation applies to a -component spino and, when the potential fo the electomagnetic field is included using the Pauli matices, allows fo the coect desciption of non-elativistic spin, including the Sten-Gelach esults. If we let ψ1 x, t Ψ ψ x, t and ϕ, A be the scala and vecto potentials of electodynamics, then the Pauli equation is i Ψ t m [σ i ea] Ψ + eϕψ fo a spin- 1 paticle with chage e. Hee, the Pauli matices ae given by σ σ x, σ y, σ z , 0 i i 0 1 0, 0 1 and the quantity [σ i ea] woks out as [σ i ea] Fom the execises we know that i σ i x i eσ ia i i σ i x i eσ ia i i σ x eσ A σ σ i x i x i σ i x i eσ A i eσ i σ A i x + e σ i σ A i A σ σ i δ i 1 + iε ik σ k Then, because ε ik ε ik while both x i x x x and A i i A A A i ae symmetic, we have emembeing that the deivatives must also act on a function, [σ i ea] Φ Φ σ σ i x i x i σ i x i eσ Φ A Φ i eσ i σ A i x + e σ i σ A i A Φ A i eσ i σ x i Φ + A Φ Φ x i i eσ i σ A i x + e A Φ A Φ i e δ i 1 + iε ik σ k x i Φ + A Φ Φ x i i e δ i 1 + iε ik σ k A i x + e A Φ Φ i e A Φ + A Φ +e σ k A Φ + A Φ i ea Φ + e A Φ σ + e A Φ Φ + e B σφ i e A Φ i ea Φ + e A Φ σ + e A Φ 5