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# lecture5 - 2.57 Nano-to-Macro Transport Processes Fall 2004...

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2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 5 Quick review of Lecture 4 1. Free particles The energy can be any values determined by the wavelength. h 2 2 Ep 2 /2 m = (/ λ ) = = k 2 ; k = 2 π / , = = h = 2 m 2 m 2. Quantum well U ENERGY AND n=1 n=2 n=3 = WAVEFUNCTION x U=0 Energy has discrete levels, and we have one quantum number n. 2 hn 2 E = 2 (n=1,2,…) 8 mD For 2D constraints, we have two quantum numbers n and l . In the discussions, the conception of “degeneracy” is introduced. 3. Spin 1 For electrons, we have talked about s = ± , where s=1/2 is called spin up and s=-1/2 2 is called spin down. 4. Harmonic oscillator 1 K = ν ( E h n + 1/ 2); = n 2 m 2.57 Fall 2004 – Lecture 5 1

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h ν Note: the zero point (intrinsic) energy is required by the Heisenberg uncertainty 2 = t <∆ ><∆ E >≥ principle ( 2 ). u x 5. Electrons moving around the nucleus e -e u r E n el =− Mc 1 2 13.6 eV ( n , 1 n A + 1 and m A , A =0, 1, 2, …) 22 2 = n n 2 E 3s 2s 1s 2p 3p 3d (- 13.6 eV) (-3.4 eV) (-1.5 eV Ψ Ψ Ψ Ψ Ψ n=1 n=2 n=3 100s 200s 21(-1)s, 210s, 21(+1)s 8 quantum states 2 quantum states ) 18 quantum states The corresponding Ψ nlms are marked in the figure for some energy levels. The degeneracy follows g(n)=2n 2 . Note: As electrons number goes up, the orbit will split. The energy of 3d is lifted up above 4s because of the electron-electron interaction. For the element potassium (K), it has 19 electrons but the n=3 energy levels are not totally filled and one electron goes to the 4s orbit. 2.57 Fall 2004 – Lecture 5 2
2.3.5 Energy Quantization observation Absorption or emission of photon happening only ( E photon ) = h ν p = E f E i E f h ν p E i E The allowable energy levels of the electron-nucleus system (hydrogen atom) are el =− 13.6 2 eV . n n The emission occurring between n=1,2 is 1 1 h 13.6 e V ( 2 2 ) ~10 e . p 1 2 The corresponding wavelength is λ = c / p . p Sometimes, we also use the wave number as 1 = p [ cm 1 ]. c p These units are used interchangeably and you should be able to do the conversion yourself. One good number to remember is that 1eV is 1.24 µm. Now we are in a position to discuss the total energy of an atom or molecule. The total energy can be approximated as the summation of translational, vibrational, rotational, and electronic energies: tot = E trans E + E el + E vib + E rot . We talk about the translational energy as a particle in a box. To simplify, for hydrogen molecules we neglect other effects and only consider the vibrational and rotational energies.

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lecture5 - 2.57 Nano-to-Macro Transport Processes Fall 2004...

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