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Coherent states and uncertainty relations
Volume 48A, number 3
PHYSICS LETTERS
17 June 1974
COHERENT STATES AND UNCERTAINTY RELATIONS D.A. TRIFONOV Institute for Nuclear Research and Nuclear Energetics, Sofia, Bulgaria Higher Paedagogical Institute, Shumen, Bulgaria Received 24 April 1974 The most general form of the minimality-preserving Hamiltonians is found and the uncertainty products ~q~p are calculated for some quadratic quantum systems
It is well known, that the uncertainty product ~qL~pequals its minimal value 1/2 in the normalizable state I~p) if Ip) is an eigenstate of the annihilation operator (1) a(j.z) = (2pi1”2(q+iiip), where ~i is an arbitrary positive number. It is clear, that for any quantum system at some fixed moment, say t = 0, one can construct a minimum uncertainty state (m.u.s.) as an eigenstaté of a(11 0) and in terms of the eigenvalue a being the m.u.s. takesand the frequency. form of a coher1, M and w the mass entAt state (c.s.) of the stationary with 11S=is(Mw)— later[1]time theIa) evolution gives Sla> oscillator a; t), where a T-exponent of the Hamiltonian and in general Ia; t) is not a m.u.s. According to the theorem a; t) will keep the minimality if it remains as eigenstate of the lowering operator a(11) for some time dependent 11(t) > 0, ji(O) = 110. Then the problem of finding the general form of the Hamiltonian, which keeps the uncertainty products minimum is similar (but not identical) to the problem of time evolution of c.s. [2] and we arrive at the following general form of the minimality-preserving Hamiltonian (2) =
2(pk(t))}+ F(t)ak(11k(t)) + Fk(t) a~Q.sk(t))+ 13(t),
w
1~~(t)a1 (,i,(t))ak(pk(t)) ÷i [~(t)/4~,~(t)] [a~(,.Lk(t))— a~ where/,k = 1,2, ... , N and ak(pk(t)) are lowering operators of the form (1) with time dependent pk(t)
> 0, j•z = dp/dt; wjk(t) = w,
1(t) and 11k(t), 13(t) are arbitrary real and Fk(t) are arbitrary complex functions of thne. Form (2) is valid both in Schradinger and Heisenberg pictures. In the one-dimensional case 20.s)) is lacking there and we observe, thatanalogous if j~= 0 forformula written to in the [3],general but theform termof (,.1/411)(a2(11)_a+ mula (2)was is identical the Hamiltonian, for which c.s. Ia) remains c.s. at all times with respect to the initial operator a(p 0). It is worth noting, that a; t) is an eigenstate of the invariant A(t) = Sa(110)S~,i.e. the exact time evolution of m.us. (and c.s.) is given by c.s. of the type, considered in [4]. This provides the possibility of using the invariants A(t) and c.s. Ia; t) (constructed in [4]) in order to calculate the evolution of the uncertainty products. As a first example we consider the general one-dimensional quadratic Hamiltonian 2+b(t)[q,p] + +c(t)q2 +d(t)p + e(t)q +f(t), (3) = a(t)p where q and p are coordinate and momentum and the coefficients are arbitrary real functions of time. The result is (~q)2(i~p)2 = 4~[1+p4(2b—jp~—ti/2a)2], where p ~+
=
(4)
Ie(t)I and e(t) is a solution of the equation
~Z~(t)e=
o,
ç~2 4ac ÷ 2b&z1+~(2a)1—4b2—2b—3t~2/4a2.
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Volume 48A, number 3
PHYSICS LETTERS
17 June 1974
From (4) and (5) we get, that Hamiltonian (3) keeps the minimality under the restriction b =(/ic—a~~)/8ac which leads to formula (2) with p = (a/c)h/2 (N 1). As a second example we consider the motion of a charged particle in a varying magnetic field H(t). If the potential A = (HX r~/2then the Hamiltonian ~C= (p— eA)2/2M is quadratic and we obtain = = 2—1(1 +~2p2)~2, (6) where p = Ie(t)I and obeys eq. (5), this time fl(t) being equal to eH(t)/2M. Products (6) keep the minimum 1/2 if p = 0 which implies constant magnetic field and this is in agreement with Hamiltonian form (2). Thus, the time dependence of the magnetic field (as well as the time dependence of the oscillator frequency w(t)) always increases the uncertainty products. The adiabatic treatment shows, that this increase is exponentially small if H(t) (or w(t)) depends analytically on time. The author is grateful to VI. Man’ko for discussion.
References [1] [2] [3] [4]
R.J. Glauber, Phys. Rev. 131 (1963) 2766. C.L. Mehta, P. Chand, E.C. Sudarshan and Vedarn, Phys. Rev. 157 (1967) 1198. D. Stoler, Phys. Rev. Dl (1970) 3217. l.A. Malkin, V.1. Man’ko and D.A. Trifonov, Phys. Lett. 30A (1969) 414; J. Math. Phys. 14 (1973) 576.
166
PHYSICS LETTERS
17 June 1974
COHERENT STATES AND UNCERTAINTY RELATIONS D.A. TRIFONOV Institute for Nuclear Research and Nuclear Energetics, Sofia, Bulgaria Higher Paedagogical Institute, Shumen, Bulgaria Received 24 April 1974 The most general form of the minimality-preserving Hamiltonians is found and the uncertainty products ~q~p are calculated for some quadratic quantum systems
It is well known, that the uncertainty product ~qL~pequals its minimal value 1/2 in the normalizable state I~p) if Ip) is an eigenstate of the annihilation operator (1) a(j.z) = (2pi1”2(q+iiip), where ~i is an arbitrary positive number. It is clear, that for any quantum system at some fixed moment, say t = 0, one can construct a minimum uncertainty state (m.u.s.) as an eigenstaté of a(11 0) and in terms of the eigenvalue a being the m.u.s. takesand the frequency. form of a coher1, M and w the mass entAt state (c.s.) of the stationary with 11S=is(Mw)— later[1]time theIa) evolution gives Sla> oscillator a; t), where a T-exponent of the Hamiltonian and in general Ia; t) is not a m.u.s. According to the theorem a; t) will keep the minimality if it remains as eigenstate of the lowering operator a(11) for some time dependent 11(t) > 0, ji(O) = 110. Then the problem of finding the general form of the Hamiltonian, which keeps the uncertainty products minimum is similar (but not identical) to the problem of time evolution of c.s. [2] and we arrive at the following general form of the minimality-preserving Hamiltonian (2) =
2(pk(t))}+ F(t)ak(11k(t)) + Fk(t) a~Q.sk(t))+ 13(t),
w
1~~(t)a1 (,i,(t))ak(pk(t)) ÷i [~(t)/4~,~(t)] [a~(,.Lk(t))— a~ where/,k = 1,2, ... , N and ak(pk(t)) are lowering operators of the form (1) with time dependent pk(t)
> 0, j•z = dp/dt; wjk(t) = w,
1(t) and 11k(t), 13(t) are arbitrary real and Fk(t) are arbitrary complex functions of thne. Form (2) is valid both in Schradinger and Heisenberg pictures. In the one-dimensional case 20.s)) is lacking there and we observe, thatanalogous if j~= 0 forformula written to in the [3],general but theform termof (,.1/411)(a2(11)_a+ mula (2)was is identical the Hamiltonian, for which c.s. Ia) remains c.s. at all times with respect to the initial operator a(p 0). It is worth noting, that a; t) is an eigenstate of the invariant A(t) = Sa(110)S~,i.e. the exact time evolution of m.us. (and c.s.) is given by c.s. of the type, considered in [4]. This provides the possibility of using the invariants A(t) and c.s. Ia; t) (constructed in [4]) in order to calculate the evolution of the uncertainty products. As a first example we consider the general one-dimensional quadratic Hamiltonian 2+b(t)[q,p] + +c(t)q2 +d(t)p + e(t)q +f(t), (3) = a(t)p where q and p are coordinate and momentum and the coefficients are arbitrary real functions of time. The result is (~q)2(i~p)2 = 4~[1+p4(2b—jp~—ti/2a)2], where p ~+
=
(4)
Ie(t)I and e(t) is a solution of the equation
~Z~(t)e=
o,
ç~2 4ac ÷ 2b&z1+~(2a)1—4b2—2b—3t~2/4a2.
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Volume 48A, number 3
PHYSICS LETTERS
17 June 1974
From (4) and (5) we get, that Hamiltonian (3) keeps the minimality under the restriction b =(/ic—a~~)/8ac which leads to formula (2) with p = (a/c)h/2 (N 1). As a second example we consider the motion of a charged particle in a varying magnetic field H(t). If the potential A = (HX r~/2then the Hamiltonian ~C= (p— eA)2/2M is quadratic and we obtain = = 2—1(1 +~2p2)~2, (6) where p = Ie(t)I and obeys eq. (5), this time fl(t) being equal to eH(t)/2M. Products (6) keep the minimum 1/2 if p = 0 which implies constant magnetic field and this is in agreement with Hamiltonian form (2). Thus, the time dependence of the magnetic field (as well as the time dependence of the oscillator frequency w(t)) always increases the uncertainty products. The adiabatic treatment shows, that this increase is exponentially small if H(t) (or w(t)) depends analytically on time. The author is grateful to VI. Man’ko for discussion.
References [1] [2] [3] [4]
R.J. Glauber, Phys. Rev. 131 (1963) 2766. C.L. Mehta, P. Chand, E.C. Sudarshan and Vedarn, Phys. Rev. 157 (1967) 1198. D. Stoler, Phys. Rev. Dl (1970) 3217. l.A. Malkin, V.1. Man’ko and D.A. Trifonov, Phys. Lett. 30A (1969) 414; J. Math. Phys. 14 (1973) 576.
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