4.10 A particle is bound in a one-dimensional potential V(x), where V(x) is symmetric, i.e., V(x) = V(−x). = −x, (a) Suppose that (x) is a solution of the Schrödinger equation with energy E. Make the change of variables y and show that (y) is also a solution of the Schrödinger equation with energy E. (b) Since the solutions of the Schrödinger equation for a fixed value of E are unique (up to multiplication by a constant), the result from part (a) implies that √(x) = c(-x), where c is an unknown constant. Use this result to show that (x) must be either even [(x) = (x)] or odd [(-x) = v(x)]. (c) For a particle bound in a one-dimensional symmetric potential, so that V(x) = V (x), show that all of the following are true: (i) * is a symmetric function, (ii) (x) = 0, (iii) (p) = 0.
4.10 A particle is bound in a one-dimensional potential V(x), where V(x) is symmetric, i.e., V(x) = V(−x). = −x, (a) Suppose that (x) is a solution of the Schrödinger equation with energy E. Make the change of variables y and show that (y) is also a solution of the Schrödinger equation with energy E. (b) Since the solutions of the Schrödinger equation for a fixed value of E are unique (up to multiplication by a constant), the result from part (a) implies that √(x) = c(-x), where c is an unknown constant. Use this result to show that (x) must be either even [(x) = (x)] or odd [(-x) = v(x)]. (c) For a particle bound in a one-dimensional symmetric potential, so that V(x) = V (x), show that all of the following are true: (i) * is a symmetric function, (ii) (x) = 0, (iii) (p) = 0.
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![4.10 A particle is bound in a one-dimensional potential V(x), where
V(x) is symmetric, i.e., V(x) = V(−x).
= −x,
(a) Suppose that (x) is a solution of the Schrödinger equation
with energy E. Make the change of variables y
and show
that (y) is also a solution of the Schrödinger equation with energy
E.
(b) Since the solutions of the Schrödinger equation for a fixed value
of E are unique (up to multiplication by a constant), the result
from part (a) implies that √(x) = c(-x), where c is an unknown
constant. Use this result to show that (x) must be either even
[(x) = (x)] or odd [(-x) = v(x)].
(c) For a particle bound in a one-dimensional symmetric potential,
so that V(x) = V (x), show that all of the following are true:
(i) * is a symmetric function,
(ii) (x) = 0,
(iii) (p)
= 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc158a850-76a9-4504-97b9-8593e0926539%2Ff1b7eb4a-a517-45c1-aec0-f213e7f7529d%2Ft47a76q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4.10 A particle is bound in a one-dimensional potential V(x), where
V(x) is symmetric, i.e., V(x) = V(−x).
= −x,
(a) Suppose that (x) is a solution of the Schrödinger equation
with energy E. Make the change of variables y
and show
that (y) is also a solution of the Schrödinger equation with energy
E.
(b) Since the solutions of the Schrödinger equation for a fixed value
of E are unique (up to multiplication by a constant), the result
from part (a) implies that √(x) = c(-x), where c is an unknown
constant. Use this result to show that (x) must be either even
[(x) = (x)] or odd [(-x) = v(x)].
(c) For a particle bound in a one-dimensional symmetric potential,
so that V(x) = V (x), show that all of the following are true:
(i) * is a symmetric function,
(ii) (x) = 0,
(iii) (p)
= 0.
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