Related questions with answers

Is the ground-state energy of a proton trapped in a one dimensional infinite potential well greater than, less than, or equal to that of an electron trapped in the same potential well?

Suppose that an electron trapped in a one-dimensional infinite well of width 250 pm is excited from its first excited state to its third excited state. (a) What energy must be transferred to the electron for this quantum jump? The electron then de-excites back to its ground state by emitting light. In the various possible ways it can do this, what are the (b) shortest, (c) second shortest, (d) longest, and (e) second longest wavelengths that can be emitted? (f ) Show the various possible ways on an energy-level diagram. If light of wavelength 29.4 nm happens to be emitted, what are the (g) longest and (h) shortest wavelength that can be emitted afterwards?

An electron is trapped in a one-dimensional infinite potential well that is $100 \mathrm{~pm}$ wide; the electron is in its ground state. What is the probability that you can detect the electron in an interval of width $\Delta x=5.0 \mathrm{~pm}$ centered at $x=$ $50 \mathrm{~pm}$ ?

Question

The ground-state energy of an electron trapped in a one-dimensional infinite potential well is 2.6 eV. What will this quantity be if the width of the potential well is doubled?

Solution

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An electron confined to an infinite potential well can exist in only certain discrete states. If the well is one-dimensional with length $L$ , the energies associated with these quantum states are:

\begin{align}E_n=\left(\dfrac{h^2}{8mL^2} \right)n^2 \end{align}

where $m$ is the electron mass and $n$ is a quantum number. Consider an electron in an infinite one-dimensional infinite potential well of length $L$ , with a ground state energy of $E_1$ , now if the length changed to $L^\prime$ , then the new ground state energy is $E_1^\prime$ , we need to find the ratio of the $L^\prime$ to $L$ , from the first equation we can see for the two ground states everything is constant except the length, so we can write,