Home /
Expert Answers /
Advanced Physics /
please-solve-the-problem-below-and-properly-comment-on-the-solutions-do-not-copy-other-solutions-pa312

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 $?x?$ and $?p_{x}?$ 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: $?_{1}(x)=2??a1??(a2x?)e_{?x_{2}/2a_{2}}.$ You may find the following integral useful $?_{??}x_{2n}e_{??x_{2}}dx=2_{n}?_{n}(2n?1)……×3×1?????forn?1.$ (a) Use the sandwich rule to write down expressions for $?x?$ and $?x_{2}?$ and hence calculate the quantity $?x$ for the state $?_{1}(x)$. You can use the properties of the functions $?(x)$ and $x$ to argue that the expectation value $?x?=0$. (b) Use the Heisenberg uncertainty principle to give a lower bound on the uncertainty of the momentum $?p_{x}$. The momentum operator can be expressed in terms of the $A$ and $A_{†}$
operators as $p?_{x}=2?a?i??(A?A_{†}).$ (c) When the particle is in an energy eigenstate, the expectation value of $p?_{x}$ vanishes, i.e. $?p_{x}?=0$. Write down an expression for $p?_{x}$ and use the orthonormality of the energy eigenfunctions $?_{n}(x)$ to argue that the terms involving $AA$ and $A_{†}A_{†}$ do not contribute to the expectation value $?p_{x}?$, for any state described by an energy eigenfunction $?_{n}(x)$. (d) Find an expression for $?p_{x}?$ and hence calculate the uncertainty $?p_{x}$ for the state $?_{1}(x)$. Is your answer compatible with part (b)?

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