ps04soln - 1 Physics 316 Solution for homework 4 Spring...

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Unformatted text preview: 1 Physics 316 Solution for homework 4 Spring 2005 Cornell University I. EXERCISE 1 Show that whenever a solution Ψ( x,t ) of the time independent Schr¨ odinger equation (hereafter S.E.) separates into a product Ψ( x,t ) = F ( x ) G ( t ) , then F(x) must satisfy the corresponding time-independent Schr¨ odinger equation for some energy E and G(t) must be proportional to e- iE/ ¯ ht . Starting from the time-dependent S.E.:- ¯ h 2 2 m ∂ 2 ∂x 2 Ψ( x,t ) + V ( x )Ψ( x,t ) = i ¯ h ∂ ∂t Ψ( x,t ) . (1.1) we substitute Ψ( x,t ) = F ( x ) G ( t ) and divide both sides by F ( x ) G ( t ) to get:- ¯ h 2 2 m 1 F ( x ) d 2 F ( x ) dx 2 + V ( x ) = i ¯ h 1 G ( t ) dG ( t ) dt (1.2) The left hand side of (2) depends only on x, while the right hand side depends only on t. The only way this can hold for all x and t is if both are equal to a constant, C. Consider the temporal part: i ¯ h 1 G ( t ) dG ( t ) dt = C (1.3) The solution to (3) is G ( t ) = G e- iC ¯ h t , where G is a constant. This describes an oscillation with frequency ω = C ¯ h . But for a de Broglie wave, ω = E ¯ h , where E is the energy. So we can identify the constant C = E . The spatial part of the S.E. is thus:- ¯ h 2 2 m d 2 F ( x ) dx 2 + V ( x ) F ( x ) = EF ( x ) . (1.4) which is the time independent S.E. II. EXERCISE 2 A particle is in a stationary state in a potential V 1 ( x ) . How do the quantized energies for V 1 ( x ) differ from those for a potential V 2 ( x ) = V 1 ( x )+ V that is larger over all x by a constant value V . Show that the spatial wave functions agree for the two potentials. Furthermore, show that the differences between energy levels agree for the two potentials....
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ps04soln - 1 Physics 316 Solution for homework 4 Spring...

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