Question

A microscopic particle with mass m is in a one-dimensional potential well. The potential energy is infinite for x < -L and for x > +L, while for -L < x < L, the potential energy is given by: U(x) = -?²x² / (mL²(L² - x²)) The particle is in a stationary state described by the wave function ?(x) = A(1 - x²/L²) for -L < x < +L and by ?(x) = 0 elsewhere. (Assume A is a positive, real constant.) (a) What is the total energy E of the system in terms of ?, m, and L? E = ?² / mL² (b) What is the normalization constant A in terms of L? A = 1/4 * sqrt(15/L) (c) What is the probability of finding the particle in the range from x = -L/4 to x = +L/4? P = How is the probability of finding the particle in this range related to its wave function? Is the wavefunction 0 or nonzero in this range? Can you use the same indefinite integral as you used in part (b)? What is the new definite integral, given the new limits of integration? What is the value of A from part (b)?

          A microscopic particle with mass m is in a one-dimensional potential well. The potential energy is infinite for x < -L and for x > +L, while for -L < x < L, the potential energy is given by:

U(x) = -?²x² / (mL²(L² - x²))

The particle is in a stationary state described by the wave function ?(x) = A(1 - x²/L²) for -L < x < +L and by ?(x) = 0 elsewhere. (Assume A is a positive, real constant.)

(a) What is the total energy E of the system in terms of ?, m, and L?
E = ?² / mL²

(b) What is the normalization constant A in terms of L?
A = 1/4 * sqrt(15/L)

(c) What is the probability of finding the particle in the range from x = -L/4 to x = +L/4?
P =
How is the probability of finding the particle in this range related to its wave function? Is the wavefunction 0 or nonzero in this range? Can you use the same indefinite integral as you used in part (b)? What is the new definite integral, given the new limits of integration? What is the value of A from part (b)?

A microscopic particle with mass m is in a one-dimensional potential well. The potential energy is infinite for x < -L and for x > +L, while for -L < x < L, the potential energy is given by:

U(x) = -?²x² / (mL²(L² - x²))

The particle is in a stationary state described by the wave function ?(x) = A(1 - x²/L²) for -L < x < +L and by ?(x) = 0 elsewhere. (Assume A is a positive, real constant.)

(a) What is the total energy E of the system in terms of ?, m, and L?
E = ?² / mL²

(b) What is the normalization constant A in terms of L?
A = 1/4 * sqrt(15/L)

(c) What is the probability of finding the particle in the range from x = -L/4 to x = +L/4?
P =
How is the probability of finding the particle in this range related to its wave function? Is the wavefunction 0 or nonzero in this range? Can you use the same indefinite integral as you used in part (b)? What is the new definite integral, given the new limits of integration? What is the value of A from part (b)?

Added by Jimmy W.

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