They can b e used to exress q, and H as follows: 8.54: Many-body henomena in condensed matter and atomic hysics Last modied: Setember 4, 3 Lecture. Coherent States. We start the course with the discussion of coherent states. These states are of interest because they rovide a method to describe on equal terms both articles and hotons a connection to classical hysics (mechanics and electrodynamics) tools for the construction of ath integral, to be discussed later The coherent states also rovide a natural entry o i n t into the method of second quantization that will be introduced in the next lecture.. Harmonic oscillator the creation and annihilation oerators Particle in a arabolic otential: m! H = q m The ground state width can b e f o u n d by minimizing energy: = ;i h@ q [q ] = i h () h m! hhi =! min () m which gives = ( h=m!) =. It will be convenient to use nondimensionalized variables q = q~, = ( h=) ~, so that the classical hase volume is rescaled by h. Thus we obtain h! H = ~ q~ ~ = ;i@ q~ [ q~ ~] = i (3) We shall study the Hamiltonian (3) below h a ving in mind the quantum-mechanical article roblem. However, later we shall nd that the quantized electromagnetic eld is also described by a set of harmonic oscillators of the form (3). The canonical creation and annihilation oerators are dened as a = ( q~ i~) a = ( q~; i~) (4) h q = a a = i a ; a (5) h! H = a a aa = h! a a (6)
The oerators a and a obey the commutation relation Proof: [a a ] = (7) aa ; a a = ((q~ i~)(q~ ; i~) ; ( q~ ; i~)(q~ i~)) = i ( ~q~ ; q~~) = (8) As a simle alication of the oerators a and a, let us reconstruct the main facts of the harmonic oscillator quantum mechanics.. The ground state j i, also called vacuum state, rovides the lowest ossible energy exectation value h jhj i = h! h ja aj i = h! ha ja i (9) which gives the condition a =, i.e., (q i) =. Let us nd the ground (vacuum) state in the q-reresentation. Using the units with the length =, i.e., q~, ~ instead of q,, w e write q (q) (q) = = = ;q ln = ;q = () This leads to a Gaussian wavefunction (q) = ;=4 ex ;q = E = h!= (). The higher energy states can be obtained from the ground state. Starting with the commutation relations, a H = ( H ; h! ) a ah = ( H h!) a () one can show that the states n(q) = (a ) n (q) are the eigenstates. Indeed, consider = a and aly the rst relation (): (H ; h!) = a H = E a = E (3) which gives E = E h! = 3 h!= and (q) = (q ; @ q ) = q ex ;q = (4) h h Subsequently, from one obtains the eigenstate (q) / (q ; ) ex (;q =) with the energy E = E h! = 5 h!=, and so on. The recursion relation n = a n;, E n = E n; h!, gives n = A ) n n (a E n = h! n (5)
where we inserted the normalization factors A n. The factors A n can b e determined from = A h(a ) n j(a ) n i = A n nh ja:::aa :::a j i (6) H = A n; h ja (a ) n; n j i (7) h! n; = A nh ja (a ) n; j i = ::: = A n h j i = A n (8) n which gives A n = ( ) ;=. The normalized oscillator eigenstates jni = (a ) n ji ji = (9) form an orthonormal comlete set of functions, roviding a basis in the oscillator Hilbert sace. The ground state ji is also known as the vacuum state. 3. The oerators a and a written as matrices in the basis of states (9) have nonzero matrix elements only between the states jni and jn i: hnja jmi = nn m hmjajni = nn m () while all other matrix elements are zero. 4. It is convenient to dene the so-called number oerator n ^ = a a which counts the numb e r of energy quanta in the QM article roblem, or the numb e r of hotons for quantized E&M eld. In the energy basis jni, t h e numb e r oerator is diagonal: njni ^ = a ajni = njni H = h! n ^ (). Denition of coherent states. The coherent states are dened as eigenstates of the oerator a: ajvi = vjvi () where v is a comlex arameter. Exanded in the energy basis (9), jvi = P jni, n= c n the coherent state can b e reconstructed from ajvi = c n njn ; i = vc njni (3) n= n= Comaring the coecients, obtain a recursion relation c n = ( v= n)cn;, leading to v n c n = c (4) The coecient c is determined from normalization = jc n j = n= n= jvj n jvj jc j = e jc j (5) 3
4 Finally, jvi = e ;jvj = v n va jni = e ;jvj = e ji (6) n= As an examle, consider the distribution of the numb e r of quanta n ^ = a a in a coherent state. Since ^njni = njni, the distribution is given by ;jvj n = jc n j = e jvj n This is a Poisson distribution with the mean n = jvj. (7).3 The quasiclassical interretation of coherent states As we shall see below, the coherent states reresent the oints of the classical hase sace (q ). This can b e conjectured most easily from their time deendence. Alying the Schrodinger equation i@ t = H to the numb e r states, we have jni(t) = e ;i(n )! t for the numb e r states. Combined with (6), this gives with jvi(t) = e ;jvj = v jni (8) n ;i(n )!t e jni = e ;i!t= jv(t)i (9) n= v(t) = e ;i!t v (3) This denes a circular trajectory in the comlex v lane, suggesting the corresondence with classical coordinate and momentum, q = c v = c v v = v iv (3) where c is a scaling factor. The relation of coherent states with the o i n ts in a classical hase sace will be claried b e l o w. Let us nd the form of a coherent state in the q-reresentation, v (q) = hqjvi. As before, we use the units in which the length =, and write v v (q) = hqjajvi = hqj (q @ q ) jvi = (q @ q ) v (q) (3) Solving the equation q = v, obtain ln = ;q = ~v q const: ~v = v (33) and, nally, v (q) = A ex ; (q ; ~v) jaj = ;=4 e ;( ~v ) = = ;=4 e ;(v ) (34)
5 with v~ = v. The robability j v (q)j has a form of a gaussian centered at q = Re(~v), which agrees with the above interretation of v as a oin t in the hase sace (with the scaling factor taking value c = ). A more detailed icture of the hase-sace density is rovided by the Wigner distribution function W (q ) = hq xj^jq ; xie ix=h dx (35) h ; where ^ is the density matrix. For a ure state (q), the density matrix in osition sace is just ^q q = (q) (q ), and the matrix element in (35) is h:::i = (q ; x) (q x) (36) The interretation of the Wigner function as a hase-sace density is suorted by the following observations. One can check that the function (35) is real and normalized to unity. Also, the coordinate and momentum distributions, obtained by integrating over the conjugate variable, are reroduced correctly. The distribution in q is W (q )d = hqj^jqi (37) which is equal to j (q)j for a ure state, while the distribution in is R W (q )dq = ::: = j ()j h where () = (q)e ;iq dq. For a coherent state jvi, the Wigner function is given by (38) jaj x; ; (q v) ~ ; (q; x;~ v) W (q ) = e e e ix dx (39) h ; jaj v v = e ;(q;~ ) e ix; ( xi~ ) ;(q;v ) ;(;~ v ) dx = e (4) h ; h with v~ = v, v~ = v. The gaussian distribution, centered at q = v~, = v~, evolves in time as if carried by the classical harmonic oscillator hase ow. Since v(t) = e ;i!t v, the center of the gaussian acket is circling around the hase sace origin: W (q t ) = e ;(q;j~ h vj cos!t ) ;(;j~vj sin!t ) For any jvi, the width of the Wigner distribution is the same as for the vacuum state ji. Thus one can conclude that a coherent state can be thought o f a s a dislaced vacuum state. This interretation will be substantiated in Problem, PS#. (4)
.4 Coherent states vector algebra Here we discuss the the vector sace roeties of coeherent states. Normally, the states aearing in quantum mechanics are orthogonal, or can b e made orthogonal in some natural way, which rovides an orthonormal basis in Hilbert sace. The situation with coherent states is quite dierent. Let us start with evaluating the overla: hujvi = e ; ; juj ; jvj uv juj e ; jvj ( uv) n n= which shows that the coherent states are not orthogonal. On the other hand, Eq.(4) gives overla decreasing exonentially as a function of the distance b e t ween u and v in the comlex lane: jhujvij = e ;ju;vj (43) For generic classical states, juj jvj, the overla is very small, which is consistent with the intuition that dierent classical states are orthogonal in the quantum mechanical sense. Recalling the interretation of the comlex v lane as a hase sace, q~ = v, ~ = v, w e see that the overla falls to zero at the length scale of the order of the waveacket width set by Planck's constant, i.e. by the uncertainty relation, q / h, h= / h. Another roerty of coherent states is comleteness in the vector algebra sense. (A set of vectors is called comlete if linear combinations of these vectors san the entire vector sace.) The roerty is seen most readily from the formula know a s u n i t y decomosition: d v jvihvj = ^ (44) Proof can be obtained by e v aluating the matrix elements of the oerator on the left hand side of Eq. (44) between the numb e r states D m E m n v v d v d v jvihvj n = e ;jvj n = e (4) r mn e i(n;m) rdrd = e ;r (45) m! ; m! r = m n e ;r dr = m n (46) i (we used olar coordinates v = re ). Using the formula (44), one can exress any oerator in terms of coherent states: ^ ^ d vd u M = ^M ^ = juihvj M (u v) (47) with the matrix elements M (u v) = hujm ^ jvi. This formula can be useful in calculations, as well as in formal maniulations (we shall use it later to derive F eynman ath integral). As another alication of Eq. (44), let show that the coherent states form an overcomlete set, i.e. they are not linearly indeendent. Indeed, by writing jvi = ^jvi = d u juihujvi = jui e ; juj ; jvj uv d u = jui e 6 uv;vu u e ; ju;vj d (48)
we exress the state jvi as a suerosition of the states jui with ju ; vj. The overcomleteness (48) should not come as a surrise. The coherent states, arameterized by comlex numbers, form a continuum, and thus there are way too many of them to form an a set of indeendent vectors. In contrast, the numb e r states, which rovide a basis of the oscillator Hilbert sace, are a countable set. To summarize, the coherent states are non-orthogonal and form an over-comlete set. There have b e e n many attemts to reduce the numb e r of these states to a `neccessary minimum,'by identifying a good subset that could serve as a basis. Even though some of the roosals are very interesting (e.g. Perelomov lattices ) it is robably more natural to use the entire sace of coherent states, coing with the overcomleteness and not favoring some of the states to the others..5 Coordinate and momentum uncertainty We already mentioned, while discussing the Wigner function, that the coherent states form waveackets in the hase sace of width corresonding to the absolute minimum required by the uncertainty relation. Let us estimate coordinate uncertainty of a state jui: hujq^jui = huj(a a ) jui = huja a a a jui = (u u) (hujq^jui) = huja a jui = (u u) h hujq^jui = hujq^ jui ; (hujq^jui) = = (49) m! The uncertainty does not deend on u, which is consistent with the observations made using Wigner function. Similarly, for momentum uncertainty, (ih) h) huj^jui = huj(a ;a) jui = (i h huja a ; a a ; jui = ; (u; u) (i h) (ih) (huj^jui) = huja ; ajui = (u ; u) h hm! huj^jui = huj^ jui ; (huj^jui) = = (5) which is also indeendent o f u. The uncertainty roduct h^i = hq^i = equals h, which is the lower bound required by the uncertainty relation. Below w e shall see that coherent states can be naturally generalized to a broader class of states that minimize uncertainty roduct. One can consider lattices in the comlex lane, v m n = mu nu, m n. Perelomov s h o wn that the lattice fv m n g generates an undercomlete set of coherent states fjv m n ig if the area of the lattice unit cell is greater than h, a n d a n o vercomlete set if the area is less than h. The borderline lattices, having the unit cell area equal to h, are overcomlete just by one vector. After any single vector is removed from such a lattice, it becomes a comlete set. 7