Fund Quantum Mechanics Lect & HW Solutions 63

Fund Quantum Mechanics Lect & HW Solutions 63 - E n...

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4.2. THE HYDROGEN ATOM 45 4.2.3.3 Solution hydc-c Question: Based on the results of the previous question, what is the color of the light emitted in a Balmer transition from energy E 3 to E 2 ? The Planck-Einstein relation says that the angular frequency ω of the emitted photon is its energy divided by ¯ h , and the wave length of light is 2 πc/ω where c is the speed of light. Typical wave lengths of visible light are: violet 400 nm, indigo 445 nm, blue 475 nm, green 510 nm, yellow 570 nm, orange 590 nm, red 650 nm. Answer: The energy carried away by the photon is the energy lost by the electron, which is E 3 E 2 = 1 . 51 eV + 3 . 4 eV = 1 . 89 eV 1 . 602 10 19 J 1 eV = 3 . 026 10 19 J Dividing by ¯ h = 1.054 10 34 J s gives the angular frequency to be 2.87 10 15 /s, and then the wave length is 656 nm, using c = 3 10 8 m/s. That will be red light. 4.2.3.4 Solution hydc-d Question: What is the color of the light emitted in a Balmer transition from an energy level
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Unformatted text preview: E n with a high value of n to E 2 ? Answer: It is like the previous question, except E n will be approximately zero at high values of n . E n E 2 = 0 eV + 3 . 4 eV = 3 . 4 eV 1 . 602 10 19 J 1 eV = 5 . 45 10 19 J Dividing by h = 1.054 10 34 J s gives the angular frequency to be 5.17 10 15 /s, and then the wave length is 364 nm. That will be near ultraviolet. The transition from E 4 will produce blue-green light, from E 5 indigo, and from E 6 violet. 4.2.4 Discussion of the eigenfunctions 4.2.4.1 Solution hydd-a Question: At what distance r from the nucleus does the square of the ground state wave function become less than one percent of its value at the nucleus? Express it both as a multiple of the Bohr radius a and in A....
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