In the potential V1(x),choosing= 0 we obtain the harmonic oscillator, where the partition function can be found in texbooks and is given by Z = x(0)=x() [dx()] exp 0 d 1 2 (dx d)2 + 1 2 2 x2(). Since the derivative of the wavefunction must give back the square of x plus a constant times the original function, the following form is suggested: Since the derivative of the wavefunction must give back the square of x plus a constant times the original function, the following form is suggested: Example 7.10 The 1D Harmonic Oscillator. The quantum harmonic oscillator has implications far beyond the simple diatomic molecule. The most accurate partition function (black line) extrapolates at low temperature to the quantum harmonic oscillator (red dashed line), at intermediate temperatures to the prediction of eq 9 (orange dotted line), and at high temperatures to the one-dimensional free translational partition function (blue dashed line). The inset shows a zoom-in . II. Partition Function for the Harmonic Oscillator . As derived in quantum mechanics, quantum harmonic oscillators have the following energy levels, E n = ( n + 1 2) where = k / m is the base frequency of the oscillator. function of the harmonic oscillator.
Such a maximum value of heat capacity in the harmonic oscillator has already been reported in the case of two-level systems , but it vanishes when q tends toward unity. The normalization factor, called canonical partition function, takes the form (still for the 1.11 Fundamentals of Ensemble Theory 29 classical uid considered in section 1.11.1) QN (T,V, N) = 1 N!h3N . only quantum statistical thermodynamics in this course, limiting ourselves to systems without interaction. Note that, even in the ground state (\(n = 0\)), the harmonic oscillator has an energy that is not zero; this energy is called the zero-point energy. The Many potentials look like a harmonic oscillator near their minimum. In this way, it was possible to compare the . 1 Introduction 7 4 Single-Quantum Oscillator 103 Singularities where Z is the partition function for the harmonic oscillator Z = 1 2sinh 2 (23) and the coecient a can be calculated [7] and has the value a = Z 12 (2n3 +3n2 + n) Path integrals in quantum mechanics, statistics, polymer physics, and financial markets | Kleinert H . Partition function of a dilute ideal gas of N particles Occupation number if two particles can occupy the same state Fluctuation in particle numbers for an . BT) partition function is called the partition function, and it is the central object in the canonical ensemble. The harmonic oscillator is an extremely important physics problem . This, however, is a totally different story and can be looked up in the authors' contributions on -function regularization in quantum field theory as published . The normalisation constant in the Boltzmann distribution is also called the partition function: where the sum is over all the microstates of the system. Thus, the partition function of the quantum harmonic oscillator is Z= e 1 2 h! H 2, Li 2, O 2, N 2, and F 2 have had terms up to n < 10 determined of Equation 5.3.1. Adding anharmonic perturbations to the harmonic oscillator (Equation 5.3.2) better describes molecular vibrations. For Boltzmann statistics, the oscillators are distinguishable and the degeneracy should be equal to the number of ways one can partition s identical objects into N different boxes, e.g. Well for a given system and reservoir, that is fixed temperature, particle number, volume or magnetic field (as appropriate), is a constant. Partition functions of boxes containing bosons or fermions Specific Heat of Diatomic Gas Rotations. Consider the one dimensional quantum harmonic oscillator with Hamiltonian H 2 = p2 T + V2 , where T is the kinetic energy (T . Anharmonic oscillation is defined as the deviation of a system from harmonic oscillation, or an oscillator not oscillating in simple harmonic . 3. In many cases we will assume that the Hamiltonian has the form H= jp~j2 2m + V(~x): (1) This de nition holds both for quantum and classical mechanics. The thd function is included in the signal processing toolbox in Matlab equation of motion for Simple harmonic oscillator 3 Isothermal Atmosphere Model 98 We have chosen the zero of energy at the state s= 0 Obviously, the effective classical potential of the cubic oscillator can be found from a variational approach only if the initial harmonic oscillator Hamiltonian has, in addition to the . Statistical Physics is the holy grail of physics. An alternative to the harmonic oscillator approximation is to include the an-harmonic effects in the partition function calculation,5-12 which is the objective of the present work. Search: Classical Harmonic Oscillator Partition Function. Then, we employ the path integral approach to the quantum non- commutative harmonic oscillator and derive the partition function of the both systems at nite temperature A limitation on the harmonic oscillator approximation is discussed as is the quantal effect in the law of corresponding states formula 32 1(1 . On page 620, the vibrational partition function using the harmonic oscillator approximation is given as q = 1 1 e h c , is 1 k T and is wave number This result was derived in brief illustration 15B.1 on page 613 using a uniform ladder. (n+ 1 2), so the harmonic oscillator partition function is given by Z . Some ideas (such as Verlinde's scenario) even place thermodynamics and statistical physics as the fundamental theory of all theories.
In order to study the anharmonic oscillator,let us sketch the solution for the single harmonic oscillator. 1.1 Partition functions This relation may be interpreted as the mean-square amplitude of a quantum harmonic oscillator 3 o ) = 2mco) h coth( /i Lorentzian distribution of the system s normal modes. The Hamiltonian is: H = [ (n k +1/2) n k] with n k =a k+ a k. Do the calculations once for bosons and once for fermions. The partition function is actually a statistial mechanics notion Except for the constant factor, Bohr-Sommerfeld quantization has done a ne job of determining the energy states of the harmonic oscillator Functional derivative and Feynman rule practice Lecture 4 - Applications of the integral formula to evaluate integrals The cartesian solution is easier and better for counting states though .
Except for the constant factor, Bohr-Sommerfeld quantization has done a ne job of determining the energy states of the harmonic oscillator., which we will review rst. This, however, is a totally different story and can be looked up in the authors' contributions on -function regularization in quantum field theory as published . In the potential V1(x),choosing= 0 we obtain the harmonic oscillator, where the partition function can be found in texbooks and is given by Z = x(0)=x() [dx()] exp 0 d 1 2 (dx d)2 + 1 2 2 x2(). in field theory. Partition function for a single particle system and for a quantum harmonic oscillator. Harmonic Oscillator and Density of States We provide a physical picture of the quantum partition function using classical mechanics in this section formula 32 1(1 The whole partition function is a product of left-movers and right-movers with some "simple adjusting factors" from the zero modes that "couple" the left-movers with the right-movers . Abstract: By harnessing quantum phenomena, quantum devices have the potential to outperform their classical counterparts. Partition Function for the Harmonic Oscillator. THE CLASSICAL PROBLEM Let m denote the mass of the oscillator and x be its displacement.
1.1 Example: Harmonic Oscillator (1D) Before we can obtain the partition for the one-dimensional harmonic oscillator, we need to nd the quantum energy levels. The Schrodinger equation with this form of potential is. The partition function for one oscillator is Q1 D Z1 1 exp p2 2m C 1 2 m!2 0x 2 dxdp h: (3) The integrations over the Gaussian functions are . Partition Function for the Harmonic Oscillator. In this article, we will work out the vibration partition function . In real systems, energy spacings are equal only for the lowest levels where the . First consider the classical harmonic oscillator: Fix the energy level =, and we may rewrite the energy relation as = 2 2 + 1 2 2 2 1= Today a modified version of their potential is used in different applications in nonlinear dynamical systems . Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. H 2 ( x) = 4 x2 - 2. (a) The Helmholtz free energy of a single harmonic oscillator is kT In(l - = -kTlnZl = - = kTln(1 - so since F is an extensive quantity, the Helmholtz free energy for N oscillators is F = NkTln(1-e ) (b) To find the entropy just differentiate with respect to T: PE) NkT(1 Nk In(l e . 3. In it I derived the partition function for a harmonic oscillator as follows q = j e j k T For the harmonic, oscillator j = (1 2 + j) for j { 0, 1, 2 The whole partition function is a product of left-movers and right-movers with some "simple adjusting factors" from the zero modes that "couple" the left-movers with the . The denominator in (4.11) (apart from normalization) is equal to the harmonic-oscillator partition function [2 sinh . 2.3.1 The harmonic oscillator partition function11 2.3.2 Perturbation theory about the harmonic oscillator partition function solution12 2.4 Problems for Section214 . Thus the partition function is easily calculated since it is a simple geometric progression, Because the system is known to exhibit periodic motion, we can again use Bohr-Sommerfeld quantization and avoid having to solve Schr odinger's equation. Therefore, you can write the wave function like this: That's a relatively easy form for a wave function, and it's all made possible by the fact that you can separate the potential into three dimensions. A quantum harmonic oscillator has an energy spectrum characterized by: where j runs over vibrational modes and It is the foundation for the understanding of complex modes of vibration in larger molecules, the motion of atoms in a solid lattice, the theory of heat capacity, etc. To recap, our answer for the equilibrium probability distribution at xed temperature is: p(fp 1;q 1g) = 1 Z e H 1(fp 1;q 1g)=(k BT) Boltzmann distribution Write down the partition function for an individual atomic harmonic oscillator, and for the collection, assuming that they have arrived in thermal equilibrium with each other at temperature T. Z S P = n = 1 e ( E n ) where is 1 / ( k B T) and the Energy levels of the quantum harmonic oscillators are E n = ( n + 1 / 2). (5)
which makes the Schrdinger Equation for . First, one can note that the system is equivalent to three independent 1D harmonic oscillators: Z 3 D = ( Z 1 D) 3 = 3 / 2 ( 1 ) 3 On the other hand, using your equation (2), we get after some algebra, is the vibrational partition function of quantum harmonic oscillator in . Partition Function for the Harmonic Oscillator . This leads to the thought that it might be possible that everything is a . It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degrees of freedom of molecules towards its thermodynamic variables. following [Benderskii et al. The Schrodinger equation with this form of potential is. The classical limits of the oscillator's motion are indicated by vertical lines, corresponding to the classical turning points at x = A of a classical particle with the same energy as the energy of a quantum oscillator in the state indicated in the figure. Partition function 1. Converged vibra-tional eigenvalue calculations have been successfully carried out for small systems such as H 2O and CH through the use of the molecular partition functions, . : Path Integral Formalism Intuitive Approach Probability Amplitude Double Slit Experiment Physical State Probability Amplitude Revisit Double Slit Experiment Distinguishability Superposition Principle Revisit the Double Slit Experiment/Superposition Principle Orthogonality Orthonormality Change of Basis Geometrical Interpretation of State . Quantum Harmonic Oscillator: Schrodinger Equation The Schrodinger equation for a harmonic oscillator may be obtained by using the classical spring potential. Dittrich, W., Reuter, M. (2020). Suppose that such an oscillator is in thermal contact with Compute the partition function Z = Tr (Exp (-H)) and then the average number of particles in a quantum state <n > for an assembly of identical simple harmonic oscillators. 1 Introduction 7 4 Single-Quantum Oscillator 103 Singularities where Z is the partition function for the harmonic oscillator Z = 1 2sinh 2 (23) and the coecient a can be calculated [7] and has the value a = Z 12 (2n3 +3n2 + n) Path integrals in quantum mechanics, statistics, polymer physics, and financial markets | Kleinert H . This is the first non-constant potential for which we will solve the Schrdinger Equation. Quantum Harmonic Oscillator: Schrodinger Equation The Schrodinger equation for a harmonic oscillator may be obtained by using the classical spring potential. If we assume the system is well-modeled by the harmonic oscillator quantum-mechanical model, the The partition function is one of the most important quantities as other thermodynamic properties can be derived from it. The partition function is actually a statistial mechanics notion The term is computed with the free particle model, as the rigid rotor and the is described as a factorization of normal modes of vibration within the harmonic oscillator In [1] they considered harmonic oscillator as a quantum system in GCE This may be shown using Stirling's . 2 Mathematical Properties of the Canonical which after a little algebra becomes This goal is, however, very Material is approximated by N identical harmonic oscillators Then, we employ the path integral approach to the quantum non- commutative harmonic oscillator and derive the partition function of the both systems at nite temperature Then . . Partition Function for the Harmonic Oscillator Walter Dittrich & Martin Reuter Chapter 2570 Accesses Part of the Graduate Texts in Physics book series (GTP) Abstract We start by making the following changes from Minkowski real time t = x 0 to Euclidean "time" = t E: \displaystyle { \tau = \text {i}t =\beta \;. } Derive the classical limit of the rotational partition function for a symmetric top molecule. Figure 81: Simple Harmonic Oscillator: Figure 82: Relative Potential Energy Minima: Expanding an arbitrary potential energy function in a Taylor series, where x 0 is the . anharmonic partition functions change with the quality of the PES in direct proportion to the harmonic-oscillator partition functions, which means frequencies (in the classical . The quantum-mechanical transition amplitude for a time-independent hamiltonian oper-ator is given by (here and henceforth we use natural units and thus set ~ = c= 1; see . MICROSTATES AND MACROSTATES From quantum mechanics follows that the states of the system do not change continuously (like in classical physics) in field theory. This is a quantum mechanical system with discrete energy levels; thus, the partition function has the form: Z = T r ( e H ^)
Such a maximum value of heat capacity in the harmonic oscillator has already been reported in the case of two-level systems , but it vanishes when q tends toward unity. The normalization factor, called canonical partition function, takes the form (still for the 1.11 Fundamentals of Ensemble Theory 29 classical uid considered in section 1.11.1) QN (T,V, N) = 1 N!h3N . only quantum statistical thermodynamics in this course, limiting ourselves to systems without interaction. Note that, even in the ground state (\(n = 0\)), the harmonic oscillator has an energy that is not zero; this energy is called the zero-point energy. The Many potentials look like a harmonic oscillator near their minimum. In this way, it was possible to compare the . 1 Introduction 7 4 Single-Quantum Oscillator 103 Singularities where Z is the partition function for the harmonic oscillator Z = 1 2sinh 2 (23) and the coecient a can be calculated [7] and has the value a = Z 12 (2n3 +3n2 + n) Path integrals in quantum mechanics, statistics, polymer physics, and financial markets | Kleinert H . Partition function of a dilute ideal gas of N particles Occupation number if two particles can occupy the same state Fluctuation in particle numbers for an . BT) partition function is called the partition function, and it is the central object in the canonical ensemble. The harmonic oscillator is an extremely important physics problem . This, however, is a totally different story and can be looked up in the authors' contributions on -function regularization in quantum field theory as published . The normalisation constant in the Boltzmann distribution is also called the partition function: where the sum is over all the microstates of the system. Thus, the partition function of the quantum harmonic oscillator is Z= e 1 2 h! H 2, Li 2, O 2, N 2, and F 2 have had terms up to n < 10 determined of Equation 5.3.1. Adding anharmonic perturbations to the harmonic oscillator (Equation 5.3.2) better describes molecular vibrations. For Boltzmann statistics, the oscillators are distinguishable and the degeneracy should be equal to the number of ways one can partition s identical objects into N different boxes, e.g. Well for a given system and reservoir, that is fixed temperature, particle number, volume or magnetic field (as appropriate), is a constant. Partition functions of boxes containing bosons or fermions Specific Heat of Diatomic Gas Rotations. Consider the one dimensional quantum harmonic oscillator with Hamiltonian H 2 = p2 T + V2 , where T is the kinetic energy (T . Anharmonic oscillation is defined as the deviation of a system from harmonic oscillation, or an oscillator not oscillating in simple harmonic . 3. In many cases we will assume that the Hamiltonian has the form H= jp~j2 2m + V(~x): (1) This de nition holds both for quantum and classical mechanics. The thd function is included in the signal processing toolbox in Matlab equation of motion for Simple harmonic oscillator 3 Isothermal Atmosphere Model 98 We have chosen the zero of energy at the state s= 0 Obviously, the effective classical potential of the cubic oscillator can be found from a variational approach only if the initial harmonic oscillator Hamiltonian has, in addition to the . Statistical Physics is the holy grail of physics. An alternative to the harmonic oscillator approximation is to include the an-harmonic effects in the partition function calculation,5-12 which is the objective of the present work. Search: Classical Harmonic Oscillator Partition Function. Then, we employ the path integral approach to the quantum non- commutative harmonic oscillator and derive the partition function of the both systems at nite temperature A limitation on the harmonic oscillator approximation is discussed as is the quantal effect in the law of corresponding states formula 32 1(1 . On page 620, the vibrational partition function using the harmonic oscillator approximation is given as q = 1 1 e h c , is 1 k T and is wave number This result was derived in brief illustration 15B.1 on page 613 using a uniform ladder. (n+ 1 2), so the harmonic oscillator partition function is given by Z . Some ideas (such as Verlinde's scenario) even place thermodynamics and statistical physics as the fundamental theory of all theories.
In order to study the anharmonic oscillator,let us sketch the solution for the single harmonic oscillator. 1.1 Partition functions This relation may be interpreted as the mean-square amplitude of a quantum harmonic oscillator 3 o ) = 2mco) h coth( /i Lorentzian distribution of the system s normal modes. The Hamiltonian is: H = [ (n k +1/2) n k] with n k =a k+ a k. Do the calculations once for bosons and once for fermions. The partition function is actually a statistial mechanics notion Except for the constant factor, Bohr-Sommerfeld quantization has done a ne job of determining the energy states of the harmonic oscillator Functional derivative and Feynman rule practice Lecture 4 - Applications of the integral formula to evaluate integrals The cartesian solution is easier and better for counting states though .
Except for the constant factor, Bohr-Sommerfeld quantization has done a ne job of determining the energy states of the harmonic oscillator., which we will review rst. This, however, is a totally different story and can be looked up in the authors' contributions on -function regularization in quantum field theory as published . In the potential V1(x),choosing= 0 we obtain the harmonic oscillator, where the partition function can be found in texbooks and is given by Z = x(0)=x() [dx()] exp 0 d 1 2 (dx d)2 + 1 2 2 x2(). in field theory. Partition function for a single particle system and for a quantum harmonic oscillator. Harmonic Oscillator and Density of States We provide a physical picture of the quantum partition function using classical mechanics in this section formula 32 1(1 The whole partition function is a product of left-movers and right-movers with some "simple adjusting factors" from the zero modes that "couple" the left-movers with the right-movers . Abstract: By harnessing quantum phenomena, quantum devices have the potential to outperform their classical counterparts. Partition Function for the Harmonic Oscillator. THE CLASSICAL PROBLEM Let m denote the mass of the oscillator and x be its displacement.
1.1 Example: Harmonic Oscillator (1D) Before we can obtain the partition for the one-dimensional harmonic oscillator, we need to nd the quantum energy levels. The Schrodinger equation with this form of potential is. The partition function for one oscillator is Q1 D Z1 1 exp p2 2m C 1 2 m!2 0x 2 dxdp h: (3) The integrations over the Gaussian functions are . Partition Function for the Harmonic Oscillator. In this article, we will work out the vibration partition function . In real systems, energy spacings are equal only for the lowest levels where the . First consider the classical harmonic oscillator: Fix the energy level =, and we may rewrite the energy relation as = 2 2 + 1 2 2 2 1= Today a modified version of their potential is used in different applications in nonlinear dynamical systems . Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. H 2 ( x) = 4 x2 - 2. (a) The Helmholtz free energy of a single harmonic oscillator is kT In(l - = -kTlnZl = - = kTln(1 - so since F is an extensive quantity, the Helmholtz free energy for N oscillators is F = NkTln(1-e ) (b) To find the entropy just differentiate with respect to T: PE) NkT(1 Nk In(l e . 3. In it I derived the partition function for a harmonic oscillator as follows q = j e j k T For the harmonic, oscillator j = (1 2 + j) for j { 0, 1, 2 The whole partition function is a product of left-movers and right-movers with some "simple adjusting factors" from the zero modes that "couple" the left-movers with the . The denominator in (4.11) (apart from normalization) is equal to the harmonic-oscillator partition function [2 sinh . 2.3.1 The harmonic oscillator partition function11 2.3.2 Perturbation theory about the harmonic oscillator partition function solution12 2.4 Problems for Section214 . Thus the partition function is easily calculated since it is a simple geometric progression, Because the system is known to exhibit periodic motion, we can again use Bohr-Sommerfeld quantization and avoid having to solve Schr odinger's equation. Therefore, you can write the wave function like this: That's a relatively easy form for a wave function, and it's all made possible by the fact that you can separate the potential into three dimensions. A quantum harmonic oscillator has an energy spectrum characterized by: where j runs over vibrational modes and It is the foundation for the understanding of complex modes of vibration in larger molecules, the motion of atoms in a solid lattice, the theory of heat capacity, etc. To recap, our answer for the equilibrium probability distribution at xed temperature is: p(fp 1;q 1g) = 1 Z e H 1(fp 1;q 1g)=(k BT) Boltzmann distribution Write down the partition function for an individual atomic harmonic oscillator, and for the collection, assuming that they have arrived in thermal equilibrium with each other at temperature T. Z S P = n = 1 e ( E n ) where is 1 / ( k B T) and the Energy levels of the quantum harmonic oscillators are E n = ( n + 1 / 2). (5)
which makes the Schrdinger Equation for . First, one can note that the system is equivalent to three independent 1D harmonic oscillators: Z 3 D = ( Z 1 D) 3 = 3 / 2 ( 1 ) 3 On the other hand, using your equation (2), we get after some algebra, is the vibrational partition function of quantum harmonic oscillator in . Partition Function for the Harmonic Oscillator . This leads to the thought that it might be possible that everything is a . It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degrees of freedom of molecules towards its thermodynamic variables. following [Benderskii et al. The Schrodinger equation with this form of potential is. The classical limits of the oscillator's motion are indicated by vertical lines, corresponding to the classical turning points at x = A of a classical particle with the same energy as the energy of a quantum oscillator in the state indicated in the figure. Partition function 1. Converged vibra-tional eigenvalue calculations have been successfully carried out for small systems such as H 2O and CH through the use of the molecular partition functions, . : Path Integral Formalism Intuitive Approach Probability Amplitude Double Slit Experiment Physical State Probability Amplitude Revisit Double Slit Experiment Distinguishability Superposition Principle Revisit the Double Slit Experiment/Superposition Principle Orthogonality Orthonormality Change of Basis Geometrical Interpretation of State . Quantum Harmonic Oscillator: Schrodinger Equation The Schrodinger equation for a harmonic oscillator may be obtained by using the classical spring potential. Dittrich, W., Reuter, M. (2020). Suppose that such an oscillator is in thermal contact with Compute the partition function Z = Tr (Exp (-H)) and then the average number of particles in a quantum state <n > for an assembly of identical simple harmonic oscillators. 1 Introduction 7 4 Single-Quantum Oscillator 103 Singularities where Z is the partition function for the harmonic oscillator Z = 1 2sinh 2 (23) and the coecient a can be calculated [7] and has the value a = Z 12 (2n3 +3n2 + n) Path integrals in quantum mechanics, statistics, polymer physics, and financial markets | Kleinert H . This is the first non-constant potential for which we will solve the Schrdinger Equation. Quantum Harmonic Oscillator: Schrodinger Equation The Schrodinger equation for a harmonic oscillator may be obtained by using the classical spring potential. If we assume the system is well-modeled by the harmonic oscillator quantum-mechanical model, the The partition function is one of the most important quantities as other thermodynamic properties can be derived from it. The partition function is actually a statistial mechanics notion The term is computed with the free particle model, as the rigid rotor and the is described as a factorization of normal modes of vibration within the harmonic oscillator In [1] they considered harmonic oscillator as a quantum system in GCE This may be shown using Stirling's . 2 Mathematical Properties of the Canonical which after a little algebra becomes This goal is, however, very Material is approximated by N identical harmonic oscillators Then, we employ the path integral approach to the quantum non- commutative harmonic oscillator and derive the partition function of the both systems at nite temperature Then . . Partition Function for the Harmonic Oscillator Walter Dittrich & Martin Reuter Chapter 2570 Accesses Part of the Graduate Texts in Physics book series (GTP) Abstract We start by making the following changes from Minkowski real time t = x 0 to Euclidean "time" = t E: \displaystyle { \tau = \text {i}t =\beta \;. } Derive the classical limit of the rotational partition function for a symmetric top molecule. Figure 81: Simple Harmonic Oscillator: Figure 82: Relative Potential Energy Minima: Expanding an arbitrary potential energy function in a Taylor series, where x 0 is the . anharmonic partition functions change with the quality of the PES in direct proportion to the harmonic-oscillator partition functions, which means frequencies (in the classical . The quantum-mechanical transition amplitude for a time-independent hamiltonian oper-ator is given by (here and henceforth we use natural units and thus set ~ = c= 1; see . MICROSTATES AND MACROSTATES From quantum mechanics follows that the states of the system do not change continuously (like in classical physics) in field theory. This is a quantum mechanical system with discrete energy levels; thus, the partition function has the form: Z = T r ( e H ^)