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EXAM3_solutions

# EXAM3_solutions - SOLUTIONS-Problems Exam3-Spring 2008 1...

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SOLUTIONS-Problems Exam3-Spring 2008 1. Suppose that a particle in the one-dimensional infinite well obeyed a rule that the quantum number n could change by only one unit as the particle make transitions among the excited states. Show that the photons emitted in those transitions have energies 3E 0 , 5 E 0 , 7 E 0 … (2n-1) E 0 … where E 0 is the ground-state energy. Solution: Consider “L” the width of the infinite well and “m” the mass of the particle. The energy of the particle confined in the one dimensional infinite well is quantized: 2 2 2 2 2 2 ± ² ³ ´ µ = = L n m m k E n n · h h . The energy of the ground state (n=1) is 0 2 2 1 2 E L m E = ± ² ³ ´ µ = · h , the energy of the n=2 state is 0 2 2 2 4 2 2 E L m E = ± ² ³ ´ µ = · h , of the n=3 state is 0 3 9 E E = . In general the energy of the “n” state is 0 2 E n E n = . Since the quantum number n can change by only one unit the energy of the photon emitted in these transitions is given by: 0 0 2 2 1 ) 1 2 ( ) ) 1 ( ( E n E n n E E n n ± = ² ± ± = ± ± , n has to be bigger than 1: For example for n=2, 0 0 1 2 3 ) 1 2 2 ( E E E E = ± ² = ± and for n=3 0 2 3 5 E E E = ± etc. 2. A particle is trapped in a one-dimensional region of dimension L . Show that the probability is 1/ n to find the particle between x=0 and x=L/n in the state of quantum number n . Solution: The wave function of a particle trapped in a one dimensional region of length L is given by ± ² ³ ´ µ · = L x n A x n ¸ ¹ sin ) ( . This wave function satisfies the Schrödinger equation and the boundary conditions 0 ) ( ) 0 ( = = L n n ± ± . The constant A can be found from the normalization condition: the probability to find the particle somewhere in confining region of length L is one. 1 2 / 2 ) ( sin ) / ( sin | ) ( | 2 2 0 2 2 0 2 2 0 2 = = = = = ± ± ± L A n n L A dy y n L A dx L x n A dx x n L L n ² ² ² ² ³ ² . Therefore L A 2 = and ± ² ³ ´ µ · = L x n L x n ¸ ¹ sin 2 ) ( . The probability to find the particle (in the state of quantum number n) located somewhere between x=0 and x=L/n is: n n L L dy y n L A dx L x n A dx x n L n L n 1 2 2 ) ( sin ) / ( sin | ) ( | 0 2 2 / 0 2 2 / 0 2 = = = = ± ± ± ² ² ² ² ³ ² . 3. According to the correspondence principle, quantum theory should give the same results as classical physics in the limit of large quantum numbers. Show that as the quantum number ± ² n , the probability of finding the trapped particle in a one-

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dimensional infinite potential well between x and x+ _ x is _ x/L and so is independent of x , which is the classical expectation. Solution. The wave function of a particle confined in a one dimensional infinite potential well is given by ± ² ³ ´ µ · = L x n L x n ¸ ¹ sin 2 ) ( . Consider a region of length _a along the x axis between x 1 =a and x 2 =a+_a. The probability to find the particle in this region is given by: L a a n L a n L a a n L a n a a a a a a n y y n dy y L n L dx L x n L dx x / ) ( / / ) ( / 2 2 2 2 2 sin 2 2 ) ( sin 2 ) / ( sin 2 | ) ( | ± + ± + ± + ± + ² ³ ´ µ · ¸ = = = ¹ ¹ ¹ º º º º º º º » ± ² ³ ´ µ + · ¸ + · ¸ + = ¹ ¸ + 2 ) / 2 sin( 2 ) / ) ( 2 sin( ) ( 1 | ) ( | 2 L a n L a n L a a n L a a n n dx x a a a n º º º º º » ( ) ) / ) ( 2 sin( ) / 2 sin( 2 1 ) ( | ) ( | 2 L a
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