By Russell L. Herman

Creation and ReviewWhat Do i must recognize From Calculus?What i would like From My Intro Physics Class?Technology and TablesAppendix: Dimensional AnalysisProblemsFree Fall and Harmonic OscillatorsFree FallFirst Order Differential EquationsThe uncomplicated Harmonic OscillatorSecond Order Linear Differential EquationsLRC CircuitsDamped OscillationsForced SystemsCauchy-Euler EquationsNumerical options of ODEsNumericalRead more...

summary: creation and ReviewWhat Do i have to recognize From Calculus?What i would like From My Intro Physics Class?Technology and TablesAppendix: Dimensional AnalysisProblemsFree Fall and Harmonic OscillatorsFree FallFirst Order Differential EquationsThe easy Harmonic OscillatorSecond Order Linear Differential EquationsLRC CircuitsDamped OscillationsForced SystemsCauchy-Euler EquationsNumerical suggestions of ODEsNumerical ApplicationsLinear SystemsProblemsLinear AlgebraFinite Dimensional Vector SpacesLinear TransformationsEigenvalue ProblemsMatrix formula of Planar SystemsApplicationsAppendix: Diagonali

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20) cos(2A) = cos2 A − sin2 A. 21) as cos(2A) = 2 cos2 A − 1, 2 = 1 − 2 sin A. Half angle formulae. 23) These, in turn, lead to the half angle formulae. Solving for cos2 A and sin2 A, we find that 1 − cos 2A , 2 1 + cos 2A cos2 A = . 2. Evaluate cos 12 . In the previous example, we used the sum/difference identities to evaluate a similar expression. We could have also used a half angle identity. In this example, we have cos2 π 12 1 π 1 + cos 2 6 √ 1 3 1+ 2 2 √ 1 2+ 3 . 26) √ π So, cos 12 = 12 2 + 3.

20) cos(2A) = cos2 A − sin2 A. 21) as cos(2A) = 2 cos2 A − 1, 2 = 1 − 2 sin A. Half angle formulae. 23) These, in turn, lead to the half angle formulae. Solving for cos2 A and sin2 A, we find that 1 − cos 2A , 2 1 + cos 2A cos2 A = . 2. Evaluate cos 12 . In the previous example, we used the sum/difference identities to evaluate a similar expression. We could have also used a half angle identity. In this example, we have cos2 π 12 1 π 1 + cos 2 6 √ 1 3 1+ 2 2 √ 1 2+ 3 . 26) √ π So, cos 12 = 12 2 + 3.

This function is the sum of terms, e− n /kT , over all possible quantum states of the system. Here, n is the energy of the nth state, T the temperature, and k is Boltzmann’s constant. Given Z, one can compute macroscopic quantities, such as the average energy, < E >= − ∂ ln Z , ∂β where β = 1/kT. For the case of the quantum harmonic oscillator, the energy states are given by = n + 12 h¯ ω, where ω is the angular frequency, h¯ is Planck’s constant divided by 2π, and n = 0, 1, 2, . . The partition function is then computed as n ∞ Z = ∑ e− β n n =0 ∞ = 1 ∑ e−β(n+ 2 )h¯ ω n =0 = e− β¯hω/2 ∞ ∑ e−βn¯hω .