Q3:2 A hamiltonian operator H has energy eigenfunctions {øn (x)} and eigenvalues {En}.The eigenfunctions {øn (x)} form an orthonormal set.(a) Write down the eigenvalue equation for the operator H. What is this equation called?!(b) The wavefunction representing a particle for this system is given by11V (x) :5ø1 (x) + ¿Ø3 (x) +594 (x).Is this wavefunction normalised to 1 ? Justify.(c) What is the probability that a measurement of the energy of the particle will yieldthe value:(i) E1 (ii) E2 (iii E4.(d) After a measurement of the energy of the particle yields the value E4, what is theprobability that a subsequent measurement of the energy of the particle will yield E1 ?

Answered: Q3:2 A hamiltonian operator H has…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Q3:2 A hamiltonian operator H has energy eigenfunctions {øn (x)} and eigenvalues {En}.
The eigenfunctions {øn (x)} form an orthonormal set.
(a) Write down the eigenvalue equation for the operator H. What is this equation called?!
(b) The wavefunction representing a particle for this system is given by
1
1
V (x) :
5ø1 (x) + ¿Ø3 (x) +594 (x).
Is this wavefunction normalised to 1 ? Justify.
(c) What is the probability that a measurement of the energy of the particle will yield
the value:
(i) E1 (ii) E2 (iii E4.
(d) After a measurement of the energy of the particle yields the value E4, what is the
probability that a subsequent measurement of the energy of the particle will yield E1 ?

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The function $f (x)$ is a rule that associates or fixes a number to the particular point x. If $y = f (x)$ , then the independent variable x is called the input of the function, and the dependent variable y is called the output of the function.

When an operator acts on a function, it transforms into another, that is, If $f (x)$ is a function and $\hat{A}$ is an operator, then $\hat{A} (x) = g (x)$ . Some of the examples for operators are $\hat{A} = \frac{d}{d x}$ , $\hat{B} = \frac{d}{d x} + \frac{d^{2}}{d x^{2}}$ etc.