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Coherent presentation of density operator of the harmonic oscillator in thermostat
Physics Letters A 372 (2007) 77–80 www.elsevier.com/locate/pla
Coherent presentation of density operator of the harmonic oscillator in thermostat R.M. Avakyan a , A.G. Hayrapetyan b,∗ , B.V. Khachatryan a , R.G. Petrosyan a a Department of Physics, Yerevan State University, 0025 Yerevan, Armenia b Institute of Applied Problems of Physics, National Academy of Sciences of Republic of Armenia, 0014 Yerevan, Armenia
Received 25 June 2007; accepted 3 July 2007 Available online 5 July 2007 Communicated by V.M. Agranovich
Abstract Based on basis of the coherent states the density matrix of harmonic oscillator in thermostat is obtained. This method is mathematically refined and physically transparent for the interpretation of quantum phenomena in classical language. Such an approach gives an opportunity to easily find the density matrix in the multi-dimensional case. © 2007 Elsevier B.V. All rights reserved. PACS: 42.25.Kb; 42.50.Ar Keywords: Coherent states in quantum theory; Condensed matter physics
It is well known that the full description of quantum systems for pure and especially for mixed ensembles is given by means of the density matrix. Naturally the problem of finding the density matrix arises. For a large number of simple model systems the given problem has a simple solution but these systems are hardly of great interest. In case of real systems lots of mathematical difficulties appear in finding density matrix. The coherent states method, which was enthusiastically developed in Glauber’s papers [1] and also by Malkin and Manko [2], is extremely fruitfully for these purposes. Integrals in this method are ordinary which makes calculations essentially easier. Furthermore they give us an opportunity to demonstrate the connection between classical and quantum theories. In present Letter one-dimensional linear oscillator in thermostat is considered. This problem was solved for the first time in [3] in the context of Schrödinger’s quantum mechanics where the mathematical apparatus is very cumbersome. The same problem is adduced also in the book [4]. Let us calculate density matrix of oscillator in coordinate and momentum representation based on the coherent states method. Hamiltonian of the linear harmonic oscillator has the following form mω2 xˆ 2 pˆ 2 + , Hˆ = 2m 2
(1)
where m is the mass, ω is the cyclic frequency, pˆ and xˆ are Hermitian operators of the momentum and coordinate respectively. Last two operators satisfy the following commutation relation [x, ˆ p] ˆ = i h. ¯ * Corresponding author.
E-mail address: [email protected] (A.G. Hayrapetyan). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.07.004
(2)
78
R.M. Avakyan et al. / Physics Letters A 372 (2007) 77–80
We introduce the annihilation and creation operators aˆ and aˆ † respectively, which are defined by the relations 1 (mωxˆ + i p), ˆ aˆ = √ 2mh¯ ω 1 (mωxˆ − i p). ˆ aˆ † = √ 2mh¯ ω
(3) (4)
Formulae (1) and (2) can be transformed in the following form 1 , Hˆ = h¯ ω aˆ † aˆ + 2 a, ˆ aˆ † = 1.
(5)
Coherent states are described by eigenfunctions of non-Hermitian operator aˆ and are defined by the following equation a|z ˆ = z|z,
(6)
where z is the corresponding eigenvalue of aˆ and 1
|z = e− 2 |z|
2
∞ zn √ |n n! n=0
(7)
is the eigenfunction. The vectors |n are the eigenfunctions of the operator of the number of quanta: aˆ † a|n ˆ = n|n,
n = 0, 1, 2, . . . .
(8)
Let us now calculate the density matrix taking into account that in quantum mechanics the state of a system in a thermostat is described by the following statistical operator ˆ
ρˆ =
e−β H , Z(β)
(9)
where Hˆ is the Hamiltonian of the system, β=
1 , kT
(10)
and ˆ
Z(β) = Tr e−β H
(11)
is the so called statistical sum. It has the following form for the linear harmonic oscillator: Z(β) =
∞
1
e
−β h¯ ω(n+ 12 )
n=0
¯ e− 2 β hω = . ¯ 1 − e−β hω
A matrix element of density operator in coordinate presentation x|ρ|x ˆ
(12)
can be presented in terms of the coherent states as 1 x|zz|ρ|z ˆ z |x d2 z d2 z . x|ρ|x ˆ = 2 π
(13)
This relation directly follows from the completeness of the coherent states and the expansion 1 |zz| d2 z 1ˆ = π 1 of the unit operator is used; d2 z = d Re z · d Im z and z = √2m (mωx + ip). h¯ ω For the calculation of the integral (13) we need to evaluate x|z and z|ρ|z ˆ . By using relation (7) for x|z one has 1
x|z = x|e− 2 |z|
2
∞ ∞ n 1 2 z zn √ |n = e− 2 |z| √ x|n. n! n! n=0 n=0
(14)
R.M. Avakyan et al. / Physics Letters A 372 (2007) 77–80
Taking into account that for the eigenfunctions of the oscillator in the coordinate representation we have 1 x 2 x x|n = ψn (x) = cn e− 2 ( b ) Hn , b √ 1 h¯ where cn = (2n n! πb)− 2 , b = mω , it is easy to verify that √ ∞ 1 2 1 x 2 1 (z/ 2)n x x|z = e− 2 |z| e− 2 ( b ) √ Hn . n! b b π n=0
79
(15)
(16)
The sum (16) is easily evaluated with the help of the generating function for the Hermite’s polynomials: ∞ n x ξ x 2 e2ξ b −ξ = Hn . n! b n=0
Finally we find
and
2
√ x 1 x √ 1 − 2 Re z x|z = √ exp −i Re z Im z + i 2 Im z − b 2 b b π
(17)
2
√ 1 1 x √ x − − 2 Re z z |x = √ exp i Re z Im z − i 2 Im z . b 2 b b π
(18)
Further let us determine z|ρ|z ˆ . Using relations (7) and (9) we have z|ρ|z ˆ =
∞
∞
e−β 2 − 1 |z|2 − 1 |z |2 z∗ n zk ¯ aˆ † aˆ |k. 2 e 2 √ √ n|e−β hω Z(β) n! k! h¯ ω
(19)
n=0 k=0
Finally the following expression is obtained h¯ ω e−β 2 1 1 exp − |z|2 − |z |2 + e−β h¯ ω z∗ z . z|ρ|z ˆ = Z(β) 2 2
(20)
ˆ = f (n)|n, where f is an analytical function. Substituting In deriving (20) we have taken into account that n|k = δnk , f (aˆ † a)|n (17), (18) and (20) in (13), presenting z and z in the following form z = Re z + i Im z ≡ q + ip and z = Re z + i Im z ≡ q + ip respectively, we obtain h¯ ω
e−β 2 1 x|ρ|x ˆ = √ 2 b π π Z(β)
−
1 2
x b
−
+∞ −∞ 2
√ 2q
2 √ x 1 x √ √ x dp dq dp dq exp −ipq + i 2p − − 2q + ip q − i 2p b 2 b b
¯ − + (q − ip)(q + ip )e−β hω
1
1 2 p + q 2 − p 2 + q 2 . 2 2
(21)
After transformations, the integral in (21) is reduced to a convenient form for further calculations: +∞
h¯ ω √ x e−β 2 1 3 3 1 1 1 x 2 1 x 2 − dp dq dp dq exp − q 2 − q 2 − p 2 − p 2 − ipq + i 2p exp − x|ρ|x ˆ = √ 2 2 2 2 b 2 b 2 2 2 2 b b π π Z(β) −∞ √ √ x x √ x ¯ + iqp e−β h¯ ω . + 2q + ip q − i 2p + 2q + qq e−β h¯ ω + pp e−β h¯ ω − ipq e−β hω (22) b b b
Let us first integrate over p , then we obtain √
+∞
h¯ ω
2 e−β 2 1 3 x 2 1 x 2 ¯ q 2 − 2q 2 − 1 1 − e−2β hω ¯ p2 − dp dq dq exp − 3 + e−2β hω exp − x|ρ|x ˆ = 2 2 2 2 b 2 b 2 2 bπ Z(β) −∞ √ √ √
2 2 2 ¯ pq + i 2 x − x e−β h¯ ω p + − i 1 − e−2β hω x + x e−β h¯ ω q + xq . b b b
(23)
80
R.M. Avakyan et al. / Physics Letters A 372 (2007) 77–80
Now, it is convenient to integrate (23) over q , after which we have +∞ √ −β h¯ ω
πe 2 1 1 x 2 1 x 2 ¯ q 2 − 1 1 − e−2β hω ¯ p 2 − i 1 − e−2β hω ¯ pq − dp dq exp − 3 + e−2β hω exp − x|ρ|x ˆ = 2 2 2 2 b 2 b 2 2 bπ Z(β) −∞ √ √
2 i 2 ¯ p+ ¯ q . + (24) x − x e−β hω x + x e−β hω b b
Integrating over p and q we obtain the well-known result derived in [3]:
2 mω mω 1 mω x|ρ|x ˆ = tanh β h¯ ω exp − x + x 2 + xx . π h¯ 2 2h¯ tanh(β h¯ ω) h¯ sinh(β h¯ ω)
(25)
Finally note that the density matrix in the momentum representation can be calculated by two ways: either from a coordinate representation with the help of Fourier-transformation or by direct calculation as it was performed above. We also have the wellknown result in the momentum representation: 1 1 p2 + p 2 pp tanh β h¯ ω exp − + . p|ρ|p ˆ = (26) πmh¯ ω 2 2mh¯ ω tanh(β hω) mh¯ ω sinh(β hω) ¯ ¯ Note, that such an approach on the evaluation of density matrix for the multi-dimensional case is efficacious by means of the complete set of commutative non-Hermitian operators, which generate all possible states from the given one. References [1] R.J. Glauber, Phys. Rev. Lett. 10 (1963) 84; R.J. Glauber, Phys. Rev. 131 (1963) 2766. [2] I.A. Malkin, V.I. Manko, Dynamical Symmetries and Coherent States of Quantum Systems, Nauka, Moscow, 1979 (in Russian). [3] F. Bloch, Zs. Phys. 74 (1932) 295. [4] L.D. Landau, E.M. Lifshitz, Statistical Physics (Part 1), Nauka, Moscow, 1976 (in Russian).
Coherent presentation of density operator of the harmonic oscillator in thermostat R.M. Avakyan a , A.G. Hayrapetyan b,∗ , B.V. Khachatryan a , R.G. Petrosyan a a Department of Physics, Yerevan State University, 0025 Yerevan, Armenia b Institute of Applied Problems of Physics, National Academy of Sciences of Republic of Armenia, 0014 Yerevan, Armenia
Received 25 June 2007; accepted 3 July 2007 Available online 5 July 2007 Communicated by V.M. Agranovich
Abstract Based on basis of the coherent states the density matrix of harmonic oscillator in thermostat is obtained. This method is mathematically refined and physically transparent for the interpretation of quantum phenomena in classical language. Such an approach gives an opportunity to easily find the density matrix in the multi-dimensional case. © 2007 Elsevier B.V. All rights reserved. PACS: 42.25.Kb; 42.50.Ar Keywords: Coherent states in quantum theory; Condensed matter physics
It is well known that the full description of quantum systems for pure and especially for mixed ensembles is given by means of the density matrix. Naturally the problem of finding the density matrix arises. For a large number of simple model systems the given problem has a simple solution but these systems are hardly of great interest. In case of real systems lots of mathematical difficulties appear in finding density matrix. The coherent states method, which was enthusiastically developed in Glauber’s papers [1] and also by Malkin and Manko [2], is extremely fruitfully for these purposes. Integrals in this method are ordinary which makes calculations essentially easier. Furthermore they give us an opportunity to demonstrate the connection between classical and quantum theories. In present Letter one-dimensional linear oscillator in thermostat is considered. This problem was solved for the first time in [3] in the context of Schrödinger’s quantum mechanics where the mathematical apparatus is very cumbersome. The same problem is adduced also in the book [4]. Let us calculate density matrix of oscillator in coordinate and momentum representation based on the coherent states method. Hamiltonian of the linear harmonic oscillator has the following form mω2 xˆ 2 pˆ 2 + , Hˆ = 2m 2
(1)
where m is the mass, ω is the cyclic frequency, pˆ and xˆ are Hermitian operators of the momentum and coordinate respectively. Last two operators satisfy the following commutation relation [x, ˆ p] ˆ = i h. ¯ * Corresponding author.
E-mail address: [email protected] (A.G. Hayrapetyan). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.07.004
(2)
78
R.M. Avakyan et al. / Physics Letters A 372 (2007) 77–80
We introduce the annihilation and creation operators aˆ and aˆ † respectively, which are defined by the relations 1 (mωxˆ + i p), ˆ aˆ = √ 2mh¯ ω 1 (mωxˆ − i p). ˆ aˆ † = √ 2mh¯ ω
(3) (4)
Formulae (1) and (2) can be transformed in the following form 1 , Hˆ = h¯ ω aˆ † aˆ + 2 a, ˆ aˆ † = 1.
(5)
Coherent states are described by eigenfunctions of non-Hermitian operator aˆ and are defined by the following equation a|z ˆ = z|z,
(6)
where z is the corresponding eigenvalue of aˆ and 1
|z = e− 2 |z|
2
∞ zn √ |n n! n=0
(7)
is the eigenfunction. The vectors |n are the eigenfunctions of the operator of the number of quanta: aˆ † a|n ˆ = n|n,
n = 0, 1, 2, . . . .
(8)
Let us now calculate the density matrix taking into account that in quantum mechanics the state of a system in a thermostat is described by the following statistical operator ˆ
ρˆ =
e−β H , Z(β)
(9)
where Hˆ is the Hamiltonian of the system, β=
1 , kT
(10)
and ˆ
Z(β) = Tr e−β H
(11)
is the so called statistical sum. It has the following form for the linear harmonic oscillator: Z(β) =
∞
1
e
−β h¯ ω(n+ 12 )
n=0
¯ e− 2 β hω = . ¯ 1 − e−β hω
A matrix element of density operator in coordinate presentation x|ρ|x ˆ
(12)
can be presented in terms of the coherent states as 1 x|zz|ρ|z ˆ z |x d2 z d2 z . x|ρ|x ˆ = 2 π
(13)
This relation directly follows from the completeness of the coherent states and the expansion 1 |zz| d2 z 1ˆ = π 1 of the unit operator is used; d2 z = d Re z · d Im z and z = √2m (mωx + ip). h¯ ω For the calculation of the integral (13) we need to evaluate x|z and z|ρ|z ˆ . By using relation (7) for x|z one has 1
x|z = x|e− 2 |z|
2
∞ ∞ n 1 2 z zn √ |n = e− 2 |z| √ x|n. n! n! n=0 n=0
(14)
R.M. Avakyan et al. / Physics Letters A 372 (2007) 77–80
Taking into account that for the eigenfunctions of the oscillator in the coordinate representation we have 1 x 2 x x|n = ψn (x) = cn e− 2 ( b ) Hn , b √ 1 h¯ where cn = (2n n! πb)− 2 , b = mω , it is easy to verify that √ ∞ 1 2 1 x 2 1 (z/ 2)n x x|z = e− 2 |z| e− 2 ( b ) √ Hn . n! b b π n=0
79
(15)
(16)
The sum (16) is easily evaluated with the help of the generating function for the Hermite’s polynomials: ∞ n x ξ x 2 e2ξ b −ξ = Hn . n! b n=0
Finally we find
and
2
√ x 1 x √ 1 − 2 Re z x|z = √ exp −i Re z Im z + i 2 Im z − b 2 b b π
(17)
2
√ 1 1 x √ x − − 2 Re z z |x = √ exp i Re z Im z − i 2 Im z . b 2 b b π
(18)
Further let us determine z|ρ|z ˆ . Using relations (7) and (9) we have z|ρ|z ˆ =
∞
∞
e−β 2 − 1 |z|2 − 1 |z |2 z∗ n zk ¯ aˆ † aˆ |k. 2 e 2 √ √ n|e−β hω Z(β) n! k! h¯ ω
(19)
n=0 k=0
Finally the following expression is obtained h¯ ω e−β 2 1 1 exp − |z|2 − |z |2 + e−β h¯ ω z∗ z . z|ρ|z ˆ = Z(β) 2 2
(20)
ˆ = f (n)|n, where f is an analytical function. Substituting In deriving (20) we have taken into account that n|k = δnk , f (aˆ † a)|n (17), (18) and (20) in (13), presenting z and z in the following form z = Re z + i Im z ≡ q + ip and z = Re z + i Im z ≡ q + ip respectively, we obtain h¯ ω
e−β 2 1 x|ρ|x ˆ = √ 2 b π π Z(β)
−
1 2
x b
−
+∞ −∞ 2
√ 2q
2 √ x 1 x √ √ x dp dq dp dq exp −ipq + i 2p − − 2q + ip q − i 2p b 2 b b
¯ − + (q − ip)(q + ip )e−β hω
1
1 2 p + q 2 − p 2 + q 2 . 2 2
(21)
After transformations, the integral in (21) is reduced to a convenient form for further calculations: +∞
h¯ ω √ x e−β 2 1 3 3 1 1 1 x 2 1 x 2 − dp dq dp dq exp − q 2 − q 2 − p 2 − p 2 − ipq + i 2p exp − x|ρ|x ˆ = √ 2 2 2 2 b 2 b 2 2 2 2 b b π π Z(β) −∞ √ √ x x √ x ¯ + iqp e−β h¯ ω . + 2q + ip q − i 2p + 2q + qq e−β h¯ ω + pp e−β h¯ ω − ipq e−β hω (22) b b b
Let us first integrate over p , then we obtain √
+∞
h¯ ω
2 e−β 2 1 3 x 2 1 x 2 ¯ q 2 − 2q 2 − 1 1 − e−2β hω ¯ p2 − dp dq dq exp − 3 + e−2β hω exp − x|ρ|x ˆ = 2 2 2 2 b 2 b 2 2 bπ Z(β) −∞ √ √ √
2 2 2 ¯ pq + i 2 x − x e−β h¯ ω p + − i 1 − e−2β hω x + x e−β h¯ ω q + xq . b b b
(23)
80
R.M. Avakyan et al. / Physics Letters A 372 (2007) 77–80
Now, it is convenient to integrate (23) over q , after which we have +∞ √ −β h¯ ω
πe 2 1 1 x 2 1 x 2 ¯ q 2 − 1 1 − e−2β hω ¯ p 2 − i 1 − e−2β hω ¯ pq − dp dq exp − 3 + e−2β hω exp − x|ρ|x ˆ = 2 2 2 2 b 2 b 2 2 bπ Z(β) −∞ √ √
2 i 2 ¯ p+ ¯ q . + (24) x − x e−β hω x + x e−β hω b b
Integrating over p and q we obtain the well-known result derived in [3]:
2 mω mω 1 mω x|ρ|x ˆ = tanh β h¯ ω exp − x + x 2 + xx . π h¯ 2 2h¯ tanh(β h¯ ω) h¯ sinh(β h¯ ω)
(25)
Finally note that the density matrix in the momentum representation can be calculated by two ways: either from a coordinate representation with the help of Fourier-transformation or by direct calculation as it was performed above. We also have the wellknown result in the momentum representation: 1 1 p2 + p 2 pp tanh β h¯ ω exp − + . p|ρ|p ˆ = (26) πmh¯ ω 2 2mh¯ ω tanh(β hω) mh¯ ω sinh(β hω) ¯ ¯ Note, that such an approach on the evaluation of density matrix for the multi-dimensional case is efficacious by means of the complete set of commutative non-Hermitian operators, which generate all possible states from the given one. References [1] R.J. Glauber, Phys. Rev. Lett. 10 (1963) 84; R.J. Glauber, Phys. Rev. 131 (1963) 2766. [2] I.A. Malkin, V.I. Manko, Dynamical Symmetries and Coherent States of Quantum Systems, Nauka, Moscow, 1979 (in Russian). [3] F. Bloch, Zs. Phys. 74 (1932) 295. [4] L.D. Landau, E.M. Lifshitz, Statistical Physics (Part 1), Nauka, Moscow, 1976 (in Russian).