Lecture 1: The superposition principle Read Baym Chapter 1, Chapter 3; Sakurai Chapter 1 1 Why do we need to learn quantum mechanics (QM)? Without quantum mechanics, we cannot understand the world. 1. Why can we stand up without collapsing? 2. Why do there exist metal/insulator/semi-conductor, magnetism, superconductivity/superfluidity? 3. What is the origin, past, future, and the fate of the earth, solar system, and even the universe? 2 Questions for further thinking 1. For a microscopic system, should it always be in a state described by a wavefunction? 2. What is the superposition principle? 3. How to normalize eigen-wavefunctions for an operator with continuous eigen-spectra? 4. How to use wavefunctions to represent probability of measurement? 5. What are the constraints for wavefunctions of identical particles? 6. Why to use linear operator to represent observable? 7. What is the precise meaning of quantization? 3 Fundamental principles and notions of QM 1. Correspondence principle 2. Space-time symmetry 3. Statistical interpretation of wavefunctions 4. Superposition principle 5. Fundamental quantization condition 6. Schodinger¨ equation for the pure state and quantum Liouville equation for the mixed state 7. Bose and Fermi statistics 1 4 How to describe classic systems? 4.1 Classic particles The state of a classical systems of N-particles can be represented by their coordinates and mo- menta as (~r1;:::;~rN;~p1;::~pN) which form a complete set of observables. Any other observable F can be expressed as a function of coordinates and momenta as F(~r1;::;~rN;~p1;:::;~pN). This is called the “purely mechanical description”. If at an initial time t0, (~r1;:::;~rN;~p1;::~pN;t0) are known, their values at any later time t are also determined by the Hamilton’s canonical equation d ¶H d ¶H qi = ; pi = − ; (i = 1;:::;n): (1) dt ¶pi dt ¶qi On the other hand, in many situations we are not able to determine all the values of coordi- nates and momenta precisely, and thus have to use the “statistical description”. If coordinates and momenta are repeatedly measured many times, a positive-definite probability distribution function is obtained r(~r1;:::;~rN;~p1;:::;~pN). The value of the observable F is expressed as R dr dr dp dp F r r p p r r p p ¯ ~1::: ~N ~ 1:: ~ N (~1;::;~N;~ 1;:::;~ N)r(~1;:::;~N;~ 1;:::;~ N) F = R : (2) d~r1:::d~rNd~p1::d~pNr(~r1;:::;~rN;~p1;:::;~pN) If the system is time-dependent, the distribution function is explicitly dependent on time as r(~r1;:::;~rN;~p1;:::;~pN;t), and its time evolution follows the Liouville’s equation ¶r N ndq ¶r dp ¶r o + ∑ i + i = 0: (3) ¶t i=1 dt ¶qi dt ¶pi 4.2 Classic waves Consider an EM wave propagating along the z-direction. Its electric fields can be written 0 0 ~ ~ as Ex(r;t) = Ex cos(kz − wt + ax), Ey(r;t) = Ey cos(kz − wt + ay), B = zˆ × E. In principle, classic physics does not need complex number. It is more convenient to represent as i(kz−wt) i(kz−wt) Ex(r;t) = Exe ; Ey(r;t) = Eye ; (4) 0 iax 0 iay where Ex = Ex e and Ey = Ey e . The polarization of the EM wave is defined as i) If Ey(x) = 0, then polarization of the EM wave is along x(y)-direction. ii) If Ey = ±iEx, then the EM wave is right(left) circularly polarized. Any general state of this EM wave can be decomposed as linear superposition of either the linear polarized bases i), or the circularly polarized basis. 2 The energy of the EM wave satisfies the following relation because 2 2 jExj + jEyj E = V = Nh¯w; (5) total 8p T where N is the number of photon. We define the state vector jYi = (yx;yy) with yx(y) = q V 2 2 8pNh¯w Ex(y), which satisfies the normalization condition jyxj + jyyj = 1. We can define a set of basis as 1 0 jxi = ; jyi = ; (6) 0 1 which are for polarizations along x and y-directions respectively. Or, we can define We can define the set of basis for circularly polarizations 1 1 1 0 jRi = p ; jLi = p : (7) 2 i 2 −i ∗ ∗ 1. Definition of a row vector hYj = (yx;yy). ∗ 2. The scalar product between jYi and jFi = (fx;fy) is defined as hYjFi = yxfx + ∗ ∗ yyfy = hFjYi . 2 2 3. Normalization condition of jyxj + jyyj = 1 can be represented as hYjYi = 1. 4. Any state can be decomposed into linear superpositions of a set of complete bases yx − iyy yx + iyy jYi = yxjxi + yyjyi = p jRi + p jLi: (8) 2 2 In other words, we have n o jYi = jxihxjYi + jyihyjYi = jxihxj + jyihyj jYi n o = jRihRjYi + jLihLjYi = jRihRj + jLihLj jYi: (9) The function of a polaroid is a projection operator. For example, the x-polaroid and y- polaroid behave as Px = jxihxj; Py = jyihyj: (10) And the polaroids of R and L-circularly polarization behave as PR = jRihRj and PL = jLihLj. For a general projection, which projects out a state jFi, its operator is represented as Pf = jFihFj. For a general state jYi, after it passes the polaroid Pf, it becomes PfjYi = hFjYijFi, where hFjYi is the amplitude passing the polaroid. The EM wave intensity after the projec- 2 2 2 tion relative to before is jPfjYij = jhFjYij , or, the probability is jhFjYij . 3 Let us perform the rotation around the z-axis at q which is denoted as R(q). Under this rotation, the bases of jxi and jyi change to jx´i = R(q)jxi and jy´i = R(q)jyi. The rotation operation R(q) can be represented as an unitary matrix as (jx´i;jy´i) = (jxi;jyi)R(q) (11) where the transformation matrix ´ Ri j(q) = hijR(q)j ji = hij j i; (12) where i; j equals to x;y, and j´ = x´;y´. More explicitly hxjx´i hxjy´i cosq −sinq R(q) = = : (13) hyjx´i hyjy´i sinq cosq Here R is an orthogonal matrix, generally speaking, it can be a unitary matrix, which can be exponentialized as − i Sq 0 −i R(q) = e h¯ ; with S = h¯ : (14) i 0 For a general state jYi, its projections on the two sets of bases satisfy hxjYi hxjYi jYi = (jxi;jyi) = (jx´i;jy´i)R−1(q) ; (15) hyjYi hyjYi and thus hx´jYi hxjYi hx´jxi hx´jyi hxjYi = R−1(q) = : (16) hy´jYi hyjYi hy´jxi hy´jyi hyjYi Define a state jY´i whose coordinates jx´i and jy´i bases are the same as those of jYi in the bases of jxi and jyi, and thus jY´i = jx´ihxjYi + jy´ihyjYi; (17) which shows that hxjY´i hx´jxi hx´jyi hxjYi hxjYi = = R(q) : (18) hyjY´i hy´jxi hy´jyi hyjYi hyjYi In other words, jY´i is the consequence of applying the rotation around z-axis at q on the state of jYi. The operator S defined in Eq. 19 is a Hermitian operator, which describe an observable: the spin angular momentum of EM wave along the z-direction. Its eigenvalues are ±h¯ with 4 eigenvecotrs YR;L 1 SY = ±h¯Y ; Y = : (19) R(L) R(L) R(L) ±i The right and left circularly polarized light carry spin (projection along the z-axis) ±h¯. 5 Description of quantum systems The quantum mechanical systems are similar to the classic waves but with significant differ- ence. For the moment, we only consider states represented by wavefunctions which obey the super-position principle. Such a principle states: 1) For a given system at a given time, the set of all the possible wavefunctions form a linear space on the complex-number field. 2) Consider a state represented by the wavefunction aYA + bYB (a, b are complex con- stants) which is a superposition of wavefunctions YA and YB. The measurements of an observable F in the state aYA + bYB only yield values from those from measurements in the states YA or YB. 3) If in the state Y, the measurement of F can yield the value of f , then Y can be decom- posed as Y = YA + YB: (20) In the state represented by YA, the measurement of F only yields the value of f ; while such measurements in the state represented by YB never yield the value of f . YA is called the eigenstate of F with the eigenvalue f . 4) Under time evolution, the linear relation between wavefunctions does not change. For example, if at t0 consider a state Y(t0) = aYA(t0) + bYB(t0), then at time t, we have Y(t) = aYA(t) + bYB(t). Examples ´ 1) For a spinless particle, we use y~r´ (~r) to represent the state of particle localized at ~r . In such a state, you can only find the particle location at ~r´, and later on we can show that ´ R ´ 3 ´ y~r´ (~r) = d(~r −~r ). Then in the state r´≤R y~r´ (~r)y(~r )d ~r , you can never find the particle outside the ball BR(0). Inside this ball, you can find the particle at the neighborhood d~r1 around the location of~r1. The corresponding component is y~r1 (~r)y(~r1)d~r1. 2) Suppose a spinless particle is located at~r0 at time t0. This wavefunction is denoted as y~r0 (~r) = d(~r −~r0). At a later time t, the amplitude of this particle located at~r is denoted as 5 K(t~r;t0~r0), which is called the propagator.