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Chapter 5: Q24E (page 187)
An electron in the n=4 state of a 5 nm wide infinite well makes a transition to the ground state, giving off energy in the form of photon. What is the photon鈥檚 wavelength?
The wavelength of the photon is .
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In the harmonic oscillators eave functions of figure there is variation in wavelength from the middle of the extremes of the classically allowed region, most noticeable in the higher-n functions. Why does it vary as it does?
For a total energy of 0, the potential energy is given in Exercise 96. (a) Given these, to what region of the x-axis would a classical particle be restricted? Is the quantum-mechanical particle similarly restricted? (b) Write an expression for the probability that the (quantum-mechanical) particle would be found in the classically forbidden region, leaving it in the form of an integral. (The integral cannot be evaluated in closed form.)
To determine the two bound state energies for the well.
What is the probability that the particle would be found between x = 0and x = 1/a?
For the harmonic oscillator potential energy, , the ground-state wave function is , and its energy is .
(a) Find the classical turning points for a particle with this energy.
(b) The Schr枚dinger equation says that and its second derivative should be of the opposite sign when E > Uand of the same sign when E < U . These two regions are divided by the classical turning points. Verify the relationship between and its second derivative for the ground-state oscillator wave function.
(Hint:Look for the inflection points.)
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