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Q.M3 - Tirgul 9 Roee Steiner Physics Department, Ben Gurion University of the Negev, BeerSheva 84105, Israel 24.12.2014 Contents 1 Gauge transformation 1.1 Classical view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Quantum view . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 3 2 Free particle states: Plane wave versus spherical waves 4 3 Classical scattering 9 1 Gauge transformation 1.1 Classical view ~ and the magnetic field B ~ can be derived from the vector The electric field E ~ potential A and the electric potential V : ~ = ∇V E ~ =∇×A ~ B (1) (2) The electric potential and the vector potential are not uniquely determined, since the electric and the magnetic fields are not affected by the following changes: V → V − φ(t) ~→A ~ + ∇λ A (3) (4) Where λ(x, t) is an arbitrary scalar function and φ(t) is scalar function that depends on time only. Such a transformation of the potentials is called ” gauge ”. A special case of gauge is changing the potential V by an addition of a constant. ∗ e-mail: [email protected] 1 ∗ Gauge transformations do not affect the classical motion of the particle since ~,B ~ the equations of motion contain only the derived fields E 1 ~ e~ ∂2x = (eE − B × ẋ) ∂t2 m c (5) This equation of motion can be derived from the Lagrangian: L= 1 e ~ mẋ2 + ẋA − e(V − φ(t)) 2 c (6) Or, alternatively, from the Hamiltonian: H= 1.2 1 e~ 2 (~ p − A) + e(V − φ(t)) 2m c (7) Quantum view From the Hamiltnian above we can see that the full Schroedinger equation (namely, with magnetic potential) is: (i~ ~ 1 ∂ ~ + eA )2 ψ(x, t) + V (x)ψ(x, t) + φ(t))ψ(x, t) = (i~∇ ∂t 2m c (8) ~ is magnetic where φ is some scalar potential that depends on time only, and A potential, namely: i~∂x + eAc x ~ ~ + eA = i~∂y + eAy i~∇ (9) c c eAz i~∂z + c The expatiation value in the quantum mechanics should be the same as in the classical case. So transformation like equations 3 should not change also the expatiation value in quantum mechanics. But unlike the case of classical mechanics (which gauge transformation dont change any measurable quantity), in quantum mechanics gauge transformation involves phase transformation of the wave function. Let see how its work: ~ + eA~ can be written as covariant derivative, namely: The term i~∇ c ~ ~ = i~∇ ~ + eA D c (10) ∂ and the term i~ ∂t + φ(t): Dt = i~ ∂ + φ(t) ∂t (11) So Schroedinger equation reads: Dt ψ(x, t) = 1 ~2 D ψ(x, t) + V (x)ψ(x, t) 2m 2 (12) As we said before the expatiation value should not change, but the wave function thus involve. So lets transform the wave function by U , and we will demands that ψ † ψ = ψ 0† ψ 0 where ψ 0 = U ψ. So: ψ † ψ = ψ 0† ψ 0 = ψ ∗ U † U ψ = ψ † ψ (13) So we can conclude that U † U = 1, so U is unitary transformation. The covariant derivative is gauge invariant, namely: ~ ψ) = U (Dψ) ~ D(U (14) Dt (U ψ) = U Dt ψ (15) What do I mean gauge invariant? I mean that under transformation the Schrdinger equation look the same.namely: 1 ~2 D (U ψ(x, t)) − Dt (U ψ(x, t)) + V (x)(U ψ(x, t)) = 0 2m (16) Will be: U 1 ~2 D ψ(x, t) − U Dt ψ(x, t) + U V (x)ψ(x, t) = 0 2m (17) 1 ~2 D ψ(x, t) − Dt ψ(x, t) + V (x)ψ(x, t)) = 0 2m (18) which is: U( So the equation of motion save its form. 1.3 Abelian case In the Abelian case H and U are scalars, so [U,H] = 0, and U can be written as U = e−iλ(t,x) (because U ∗ U = 1), where λ(t, x) is some scalar function. So: ~ ~ ψ) = D(e ~ −iλ(t,x) ψ) = (i~∇ ~ + eA )(e−iλ(t,x) ψ) D(U c ~ eA ~ + ~∇λ(t, ~ = e−iλ(t,x) (i~∇ x) + )ψ c (19) We can see that if we want that ~ ψ) = U (Dψ) ~ D(U (20) Dt (U ψ) = U Dt ψ (21) ~ will transform also by we must have that A ~→A ~ − c~ ∇λ(t, ~ A x) e 3 (22) The same is true for φ φ → φ − ~∂t λ(t, x) (23) We can see that it is transforming like in equations 3 Example ~ = constant, so We have system with zero magnetic field. This means that A c~ ~ ~ lets say that A = − e ∇λ(t, x) = (0, c0 , 0) in cylinder coordinate. So ∆λ = − 2πe rc0 c~ (24) So the wave function get a phase. 2 Free particle states: Plane wave versus spherical waves The following pages of this section is from Sakurai, which cover the topic in very good way. 4 6.4 405 Phase Shi fts and Partia l Waves free-particle Hamiltonian also commutes with L2 and Lz . Thus it is possible to consider a simultaneous eigenket of Ho,L2 , and Lz . Ignoring spin, such a state is denoted by I E, l, m} and is often called a spherical-wave state. More generally, the most general free-particle state can be regarded as a super position of I E, l, m} with various E, l, and m in much the same way as the most general free-particle state can be regarded as a superposition of l k} with different k, different in both magnitude and direction. Put in another way, a free-particle state can be analyzed using either the plane-wave basis { l k} } or the spherical-wave basis { I E, l, m} } . We now derive the transformation function (kl E, l , m} that connects the plane wave basis with the spherical-wave basis. We can also regard this quantity as the momentum-space wave function for the spherical wave characterized by E, l, and m. We adopt the normalization convention for the spherical-wave eigenket as follows: (E' , l ' , m ' I E, l , m} = 8ll'8mm'8(E- E'). (6.4. 1 ) In analogy with the position-space wave function, we may guess the angular dependence: (ki E, l , m} = gzE(k)Yt (k ), (6.4.2) where the function gzE(k) will be considered later. To prove this rigorously, we proceed as follows. First, consider the momentum eigenket lkZ}-that is, a plane wave state whose propagation direction is along the positive z-axis. An important property of this state is that it has no orbital angular-momentum component in the z-direction: Lzl kZ} = (xpy- YPx) lkx = O, ky = O, kz = k} = 0. (6.4.3) Actually this is plausible from classical considerations: The angular-momentum component must vanish in the direction of propagation because L p = (x x p) p = 0. Because of (6.4.3)-and since (E ' , l' , m ' lkz} = 0 for m ' =f:. O-we must be able to expand lkz} as follows: · lkZ} = LJ l' dE ' I E' , l ' , m ' = O} (E ' , l ' , m ' = O lkz} . · (6.4.4) Notice that there is no m ' sum; m ' is always zero. We can obtain the most general momentum eigenket, with the direction of k specified by() and c/J, from lkZ} by just applying the appropriate rotation operator as follows [see Figure 3.3 and (3 .6.47)] : l k} = :D(a = cp, {J = e, y = O)l kZ} . (6.4.5) 406 Chapter 6 Scattering Theory Multiplying this equation by (E, l , m l on the left, we obtain (E, Z, mlk) = Lf l' x = (E' , l ' , m ' = Olkz) L:j l' x dE ' (E , l , m i:D (a = ¢, {3 = e, y = O)iE' , l ' , m' = 0 ) �6 (a = ¢, f3 = 8, y = 0) (6.4.6) dE':D �� (a = ¢, {3 = e, y = O) (E, l , m = OlkZ) . 8u,8(E - E' ) ( E' , l ' , m ' = OlkZ) = :D ..[iii!-g1E(k). So we can write, using Now (E, l , m = O l kZ ) is independent of the orientation of k-that is, independent of e and ¢-and we may as well call it (3 .6.5 1), (kiE , l , m ) = 8ZE (k) Yt (k). (6.4.7) Let us determine gzE(k). First, we note that (Ho - E ) I E , l , m ) = 0. (6.4.8) But we also let Ho - E operate on a momentum eigenbra (kl as follows: (6.4.9) Multiplying (6.4.9) with I E, l, m ) on the right, we obtain (6.4. 1 0) This means that (kl E, l, m) can be nonvanishing only if E = 1i2 k2 j2m, so we must be able to write gzE(k ) as (6 . 4. 1 1 ) 6.4 407 Phase S h ifts and Partia l Waves N (E',l'm'IE,l,m) I d3k"(E',l',m'lk")(k"IE,l,m) j k"2dk" j dQ.,,INI20 ( E') 0 ( -E) Y?" (k")Yt(k") (Ji�-E 2kfl2 ') (Ji�-I k"2dE" 2kfl2 E) 2 d NI Q k"I - dE"/dk" I To determine we go back to our normalization convention (6.4. 1 ). We obtain = = x x = 8 Y{;t'* (k")Yt (k") 8 INI2 mk' Ji2 8(E-E' E" 1i2k"2 m )8u'8mm'• Nk"- 1i rmf where we have defined = j2 to change integration. Comparing this with (6.4. 1 ), we see that Therefore, we can finally write = (6.4 . 1 2) E" integration into j will suffice. (6.4. 1 3) hence 2k2 E) Y1 (k). (kiE,l,m) 1i (1im lk) lk) LL I dEIE,l,m)(E,l,mlk) (�yt*ck:)). f t IE,Z,m) = 2 - m A (6.4. 14) From (6.4. 14) we infer that the plane-wave state can be expressed as a super position of free spherical-wave states with all possible !-values; in particular, = l m (6.4. 15) = l=Om=-1 E=fi2k2j2m Because the transverse dimension of the plane wave is infinite, we expect that the plane wave must contain all possible values of impact parameter b (semiclassi cally, the impact parameter b:::::: j p ). From this point of view it is no surprise when analyzed in terms of spherical-wave that the momentum eigenstates states, contain all possible values of We have derived the wave function for in momentum space. Next, we consider the corresponding wave function in position space. From wave mechan ics, the reader should be familiar with the fact that the wave function for a free llk), h l. IE,l,m) 408 Chapter 6 Scattering Theory spherical wave isjz(kr) Yt(r), where Jz(kr) is the spherical Bessel function of or der [see (3.7.20a) and also Appendix B]. The second solution nt(kr) , although it satisfies the appropriate differential equation, is inadmissible because it is singular at the origin. Thus we can write l (xiE,l,m) = cziz(kr)YtCn (6.4. 1 6) To determine q , all we have to do is compare LL / dE(xl E ,l, m )(E,l,ml k) l h ( h2mk2 ) Y1 (k) dE czjz(kr)Y1 (r) � 8 E- 2 (xlk) = (2rr) = m* m = = m A L1 (214+rr 1 ) Pz(k · r) A A (6.4. 1 7) h where we have used the addition theorem in the last step. Now (xl k) = e ik·x /(2rr) 312 can also be written as (6.4. 18) which can be proved by using the following integral representation for jz(kr): jz(kr) = 1 . 2l 1 - 1+1 . -l ezkr cose P cos z( e ) d( os ) . c e (6.4. 19) Comparing (6.4. 1 7) with (6.4 . 1 8), we have cz = iz h (6.4.20) To summarize, we have h ( h22mk2 ) Y1 (k) (kiE,l, m ) = � 8 Em (xiE,l, m ) = m jz(kr)Y1 (r). A (6.4.21a) (6.4.21b) These expressions are extremely useful in developing the partial-wave expansion. We conclude this section by applying (6.4.21a) to a decay process. Suppose a parent particle of spin j disintegrates into two spin-zero particles: A (spin j) --+ 3 Classical scattering I took the following pages from the book ”Mechanics from Newton’s laws to Deterministic Chaos” of ”Florian Scheck”. The aim is to provide Introduction to the scattering in quantum mechanics. 9 10 Before moving into the solution lates remember the solution of two body system with central force. 11 12 So 13 14