Problem 2.44 If two (or more) distinct6 solutions to the (time-independent) Schrödinger equation have the same energy E, these states are said to be degenerate. For example, the free particle states are doubly degenerate-one solution representing motion to the right, and the other motion to the left. But we have never encountered normalizable degenerate solutions, and this is accident. Prove the following theorem: In dimension 7 no опе (-00
Problem 2.44 If two (or more) distinct6 solutions to the (time-independent) Schrödinger equation have the same energy E, these states are said to be degenerate. For example, the free particle states are doubly degenerate-one solution representing motion to the right, and the other motion to the left. But we have never encountered normalizable degenerate solutions, and this is accident. Prove the following theorem: In dimension 7 no опе (-00
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![Problem 2.44 If two (or more) distinct6 solutions to the (time-independent)
Schrödinger equation have the same energy E, these states are said to be
degenerate. For example, the free particle states are doubly degenerate-one
solution representing motion to the right, and the other motion to the left.
But we have never encountered normalizable degenerate solutions, and this is
accident.
Prove
the following theorem: In
dimension 7
no
опе
(-00<x < 0) there are no degenerate bound states. [Hint: Suppose there are
two solutions, y and ý2, with the same energy E. Multiply the Schrödinger
equation for y1 by y2, and the Schrödinger equation for 2 by V1, and
subtract, to show that (2dy1/dx – Vid2/dx) is a constant. Use the fact
that for normalizable solutions y →0 at ±o to demonstrate that this
constant is in fact zero. Conclude that 2 is a multiple of , and hence that
the two solutions are not distinct.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fde1132a6-9684-48be-960f-3f9ee5658757%2F6809cb99-aa1b-432e-aeff-df904363c0fa%2Fb58u0u_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 2.44 If two (or more) distinct6 solutions to the (time-independent)
Schrödinger equation have the same energy E, these states are said to be
degenerate. For example, the free particle states are doubly degenerate-one
solution representing motion to the right, and the other motion to the left.
But we have never encountered normalizable degenerate solutions, and this is
accident.
Prove
the following theorem: In
dimension 7
no
опе
(-00<x < 0) there are no degenerate bound states. [Hint: Suppose there are
two solutions, y and ý2, with the same energy E. Multiply the Schrödinger
equation for y1 by y2, and the Schrödinger equation for 2 by V1, and
subtract, to show that (2dy1/dx – Vid2/dx) is a constant. Use the fact
that for normalizable solutions y →0 at ±o to demonstrate that this
constant is in fact zero. Conclude that 2 is a multiple of , and hence that
the two solutions are not distinct.]
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