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(Solved): Please solve the problem below and properly comment on the solutions (do not copy other solutions). ...



Please solve the problem below and properly comment on the solutions (do not copy other solutions).

???????This question relates mainly to Units 5 and 6 .
In this question you will evaluate expectation values \( \langle x\rangle \)

operators as
\[
\widehat{\mathrm{p}}_{x}=\frac{-\mathrm{i} \hbar}{\sqrt{2} a}\left(\widehat{\mathrm{A}}-\widehat{\mathrm{A}}^

This question relates mainly to Units 5 and 6 . In this question you will evaluate expectation values and for the simple harmonic oscillator using two different, but complementary techniques. A particle subject to a simple harmonic oscillator potential is prepared in the first excited energy eigenstate, which is described by the following wave function: You may find the following integral useful (a) Use the sandwich rule to write down expressions for and and hence calculate the quantity for the state . You can use the properties of the functions and to argue that the expectation value . (b) Use the Heisenberg uncertainty principle to give a lower bound on the uncertainty of the momentum . The momentum operator can be expressed in terms of the and operators as (c) When the particle is in an energy eigenstate, the expectation value of vanishes, i.e. . Write down an expression for and use the orthonormality of the energy eigenfunctions to argue that the terms involving and do not contribute to the expectation value , for any state described by an energy eigenfunction . (d) Find an expression for and hence calculate the uncertainty for the state . Is your answer compatible with part (b)?


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(a) Calculating expectation values for x and x2for first excited energy eigenstate (?1) of quantum harmonic oscillation.= informally refe
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