Lecture 42 Quantum Statistics

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1 Lecture 42 Quantum Statistics
Energy is not continuous in quantum mechanics Boltzmann factor Atoms in ground state Specific heat of quantum gas, rotation and vibration

2 Quantum mechanics In the microscopic world, the usual Newton Mechanics does not work, we need a new form of mechanics, called quantum mechanics. Two important consequences in quantum mechanics Energy is not continuous Particles are indistinguishable (fermi and bose gases) Here we study the consequence when the energy is not continuous

3 Discrete energy level In quantum mechanics, the energy of a system 𝐸 is not continuous. It takes a series of discrete values, such as 𝐸1, 𝐸2, 𝐸3, …. Thus the Boltzmann factor for each energy level is 𝑃 𝑖 ∼ 𝑒 − 𝐸 𝑖 𝑘 𝐵 𝑇 Therefore, instead of a continuous distribution in E, the probability is discrete.

4 When is quantum effect important?
Depends on when the discreteness is important, this in turn depends on the relative size between 𝑘 𝐵 𝑇 and 𝐸 𝑖 − 𝐸 𝑖+1 . The Boltzmann constant can be expressed in eV 𝑘 𝐵 =0.86× 10 −4 𝑒𝑉/𝐾 Thus at room temp 300 K , we have 𝑘 𝐵 𝑇=2.5× 10 −2 𝑒𝑉=25 𝑚𝑒𝑉

5 Atoms are in the ground state
The excitation energy of a typical atom is 𝑒𝑉, which is much large than 25𝑚𝑒𝑉 Thus at the room temp, almost all atoms are in the ground (lowest energy ) state. The probability at the first excited state is 𝑃 1 = 𝑒 − 𝐸 1 − 𝐸 0 𝑘 𝐵 𝑇 hydrogen atom, Δ𝐸=10.2 𝑒𝑉, Δ𝐸 𝑘𝑇 =400 𝑃 1 = 𝑒 −400 ~ 0

6 When quantum effect becomes less important?
In the case of atom, we need to have a temp about 1𝑒𝑉 to 10𝑒𝑉. This corresponding to 10, 000 to 100, 000K In this case, the atom can be ionized , one is dealing with a plasma of positive and negative charges.

7 Chemistry and Biology In chemistry, we are dealing with binding energy of different atoms. This biding energy usually has a range in terms of 𝑒𝑉. It can go from 𝑚𝑒𝑉 to 𝑒𝑉. In biology, we are dealing with arrangement of large molecules, cells, etc, the energy scale is even lower. Biology is very temp-sensitive.

8 An example: Harmonic oscillator
For the quantum harmonic oscillator, the energy is discontinuous, 𝐸 𝑛 =ℏ𝜔 𝑛 , 𝑤ℎ𝑒𝑟𝑒 𝑛=0, 1,2, 3…. The lowest energy is not zero, but ℏ𝜔 2 ， this is called zero-point energy, coming from quantum mechanical uncertainty principle. The probability is 𝑃 𝑛 ∼ 𝑒 −𝑛 ℏ𝜔 𝑘 𝐵 𝑇

9 Average energy Then the energy becomes 𝐸=−𝜕𝑍/𝜕𝛽 The average energy
𝐸 = 𝑛 𝐸 𝑛 𝑃 𝑛 Introduce the partition function, and 𝛽=1/𝑘𝑇 𝑍= 𝑛 𝑒 − 𝐸 𝑛 𝑘𝑇 Then the energy becomes 𝐸=−𝜕𝑍/𝜕𝛽

10 Specific heat We can calculate the specific heat 𝐶 𝑉 = 𝜕𝐸 𝜕𝑇
At high-temp, it becomes, 𝑘𝑇 At low-temp, it goes to zero.

11 Monatomic and diatomic gases
Monatomic gases have only translational motion, 3 degrees of freedom 𝑈 𝑚𝑜𝑙 = 3 2 𝑁 𝐴 𝑘 𝐵 𝑇= 3 2 𝑅𝑇 Diatomic gases, 3 degrees of freedom for translational motion the center of mass, 2 degrees of freedom for rotation, 1 degree of freedom for oscillation and 1 degree of freedom for potential 𝑈 𝑚𝑜𝑙 = 7 2 𝑁 𝐴 𝑘 𝐵 𝑇= 7 2 𝑅𝑇

12 Constant volume specific heat capacity of a diatomic gas
As temperature increases, heat capacity goes from 3 2 𝑅 (translation contribution only), to 5 2 𝑅 (translation + rotation), finally to a maximum of 7 2 𝑅 (translation + rotation + vibration)