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Q2 Consider a particle moving in one dimension and described by the following Hamil- tonian operator, H +x?. dx? (a) Verify that (x) = Ar exp is an eigenfunction of H and determine the associated eigenvalue. (b) Determine the value of A so that the norm of , | = V(, b) = 1 (i.e. the wavefunction is normalised to unity). %3D (Hint: Integrate by parts and use the result exp (-x²) dx = T.) (c) If the particle's quantum state is represented by v, write down an integral expression for the probability of finding the particle somewhere in the interval -1

Q2 Consider a particle moving in one dimension and described by the following Hamil- tonian operator, H +x?. dx? (a) Verify that (x) = Ar exp is an eigenfunction of H and determine the associated eigenvalue. (b) Determine the value of A so that the norm of , | = V(, b) = 1 (i.e. the wavefunction is normalised to unity). %3D (Hint: Integrate by parts and use the result exp (-x²) dx = T.) (c) If the particle's quantum state is represented by v, write down an integral expression for the probability of finding the particle somewhere in the interval -1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Q2 Consider a particle moving in one dimension and described by the following Hamil- tonian operator, H +x?. dx? (a) Verify that (x) = Ax exp | is an eigenfunction of H and determine the associated eigenvalue. (b) Determine the value of A so that the norm of , | = V(, b) = 1 (i.e. the wavefunction is normalised to unity). %3D (Hint: Integrate by parts and use the result exp (-x²) dx = T.) (c) If the particle's quantum state is represented by v, write down an integral expression for the probability of finding the particle somewhere in the interval -1 <x < 2. %3D
Transcribed Image Text:Q2 Consider a particle moving in one dimension and described by the following Hamil- tonian operator, H +x?. dx? (a) Verify that (x) = Ax exp | is an eigenfunction of H and determine the associated eigenvalue. (b) Determine the value of A so that the norm of , | = V(, b) = 1 (i.e. the wavefunction is normalised to unity). %3D (Hint: Integrate by parts and use the result exp (-x²) dx = T.) (c) If the particle's quantum state is represented by v, write down an integral expression for the probability of finding the particle somewhere in the interval -1 <x < 2. %3D
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Introduction

If f is a eigenfunction with response to the operator H , then :  H[f] = λf , where λ is the corresponding eigenvalue.

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