Question

Consider a particle of mass $m$ trapped in an infinitely deep one-dimensional potential well located between $x = -a$ and $x = a$. Experimentalists perform an extensive range of accurate experiments and find (after av- eraging over multiple experiments) that there is a 10% probability for the particle to be located between $x = 0$ and $x = a/5$, i.e. $P(0 \le x \le \frac{a}{5}) = \frac{1}{10}$. (1) (a) By considering the shape (or number of nodes) of the wavefunctions of possible eigenstates of the system (or otherwise), deduce an ex- pression for the corresponding normalised eigenstate of the system, and deduce its corresponding energy. (b) Is such a result unique, or are there other (different energy) eigen- states satisfying Eq. (1)? (c) Using your obtained eigenfunction, explicitly verify your result by direct calculation of $P(0 \le x \le a/5)$ through the relevant integral.

          Consider a particle of mass $m$ trapped in an infinitely deep one-dimensional
potential well located between $x = -a$ and $x = a$. Experimentalists
perform an extensive range of accurate experiments and find (after av-
eraging over multiple experiments) that there is a 10% probability for
the particle to be located between $x = 0$ and $x = a/5$, i.e.
$P(0 \le x \le \frac{a}{5}) = \frac{1}{10}$.
(1)
(a) By considering the shape (or number of nodes) of the wavefunctions
of possible eigenstates of the system (or otherwise), deduce an ex-
pression for the corresponding normalised eigenstate of the system,
and deduce its corresponding energy.
(b) Is such a result unique, or are there other (different energy) eigen-
states satisfying Eq. (1)?
(c) Using your obtained eigenfunction, explicitly verify your result by
direct calculation of $P(0 \le x \le a/5)$ through the relevant integral.

Consider a particle of mass m trapped in an infinitely deep one-dimensional
potential well located between x = -a and x = a. Experimentalists
perform an extensive range of accurate experiments and find (after av-
eraging over multiple experiments) that there is a 10% probability for
the particle to be located between x = 0 and x = a/5, i.e.
P(0 ≤ x ≤(a)/(5)) = (1)/(10).
(1)
(a) By considering the shape (or number of nodes) of the wavefunctions
of possible eigenstates of the system (or otherwise), deduce an ex-
pression for the corresponding normalised eigenstate of the system,
and deduce its corresponding energy.
(b) Is such a result unique, or are there other (different energy) eigen-
states satisfying Eq. (1)?
(c) Using your obtained eigenfunction, explicitly verify your result by
direct calculation of P(0 ≤ x ≤ a/5) through the relevant integral.

Added by Joseph A.

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