Suppose the barrier height above the electron energy (U barrier -E electron ) is 1 eV and that t

step 1 Our question says to use the uncertainty principle to figure out the minimum time interval, the

Suppose the barrier height above the electron energy (U...

Suppose the barrier height above the electron energy $\left(U_{\text { barrier }}-E_{\text { electron }}\right)$ is 1 eV and that the barrier is 2 nm wide. According to the uncertainty principle, what is the minimum time interval that the electron's energy is in this classically forbidden region? $$ \begin{array}{ll}{\text { (a) } 3 \times 10^{-16} \mathrm{s}} & {\text { (b) } 3 \times 10^{-15} \mathrm{s}} \\ {\text { (c) } 3 \times 10^{-14} \mathrm{s}} & {\text { (d) } 3 \times 10^{-13} \mathrm{s}}\end{array} $$ $$ \text { (e) } 3 \times 10^{-12} \mathrm{s} $$

Suppose the barrier height above the electron energy $\left(U_{\text { barrier }}-E_{\text { electron }}\right)$ is 1 eV and that the barrier is 2 nm wide. According to the uncertainty principle, what is the minimum time interval that the electron's energy is in this classically forbidden region?
$$
\begin{array}{ll}{\text { (a) } 3 \times 10^{-16} \mathrm{s}} & {\text { (b) } 3 \times 10^{-15} \mathrm{s}} \\ {\text { (c) } 3 \times 10^{-14} \mathrm{s}} & {\text { (d) } 3 \times 10^{-13} \mathrm{s}}\end{array}
$$
$$
\text { (e) } 3 \times 10^{-12} \mathrm{s}
$$

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Key Concepts

Heisenberg Uncertainty Principle (Energy-Time Relation)

This principle in quantum mechanics states that certain pairs of physical properties, such as energy and time, cannot be simultaneously measured to arbitrary precision. Specifically, the energy-time uncertainty relation indicates that the uncertainty in energy multiplied by the uncertainty in time is on the order of Planck’s constant. In contexts like barrier penetration, this relationship suggests that an electron can temporarily “borrow” energy, allowing it to exist in a state where it classically would not, as long as it does so for a very short period.

Quantum Tunneling

Quantum tunneling is a phenomenon where particles have a finite probability to penetrate through a potential barrier even when their energy is less than the height of the barrier. This effect is a direct consequence of their wave-like nature and the inherent uncertainties described by quantum mechanics. In regions deemed classically forbidden, such as under a potential barrier, the particle’s presence is enabled by the uncertainty in its energy and temporal localization, permitting a brief existence in the barrier region.

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