Question

(a) Use the Heisenberg uncertainty principle to calculate the uncertainty in energy for a corresponding time interval of\({\rm{1}}{{\rm{0}}^{{\rm{ - 43}}}}{\rm{ s}}\). (b) Compare this energy with the\({\rm{1}}{{\rm{0}}^{{\rm{19}}}}{\rm{ GeV}}\)unification-of-forces energy and discuss why they are similar.

Short answer

1. The uncertainty in energy is obtained is \(3 \times {10^{18}}\,{\rm{GeV}}\).
2. As, the Heisenberg uncertainty principle expects a particle's existence to be too brief, this theoretical energy threshold means that such a particle could not be seen.

Step-by-step solution

Heisenberg Principle

The Heisenberg uncertainty principle is given by,

\(\Delta E = \frac{h}{{4\pi \Delta t}}\)

Here\(\Delta E\)is the uncertainty in energy,\(\Delta t\)is uncertainty in time and\(h\)is the Planck’s constant.

 Step 2: Evaluating the uncertainty in energy

(a)

Heisenberg uncertainty principle for energy states that:

\(\Delta E \cdot \Delta t \approx \frac{h}{{4\pi }}\)

Substitute\({\rm{1}}{{\rm{0}}^{{\rm{ - 43}}}}{\rm{s}}\)for\(\Delta t\)in the above equation,

So, we obtain:

\(\begin{align}\Delta E \approx \frac{h}{{4\pi \Delta t}}\\ &= \frac{{{\rm{6}}{\rm{.626 \times 1}}{{\rm{0}}^{{\rm{ - 43}}}}{\rm{ Js}}}}{{{\rm{4\pi \times 1}}{{\rm{0}}^{{\rm{ - 43}}}}{\rm{ s}}}}\\ &= {\rm{5}}{\rm{.27 \times 1}}{{\rm{0}}^{\rm{8}}}{\rm{J}}\end{align}\)

Therefore, the uncertainty is\({\rm{5}}{\rm{.27 \times 1}}{{\rm{0}}^{\rm{8}}}{\rm{ J}}\).

Convert it into \({\rm{eV}}\)

\(\begin{align}\Delta E &= \left( {5.2728 \times {{10}^8}\,{\rm{J}}} \right)\left( {\frac{{1\,{\rm{eV}}}}{{1.6 \times {{10}^{ - 19}}\,{\rm{J}}}}} \right)\\ &= 3 \times {10^{27}}\,{\rm{eV}}\\ &= 3 \times {10^{18}}\,{\rm{GeV}}\end{align}\)

Therefore the uncertainty in energy is \(3 \times {10^{18}}\,{\rm{GeV}}\).

Explanation for part b

(b)

The ratio is

\(\begin{align}\frac{{\Delta E}}{E} &= \frac{{3 \times {{10}^{18}}\,{\rm{GeV}}}}{{{{10}^{18}}\,{\rm{GeV}}}}\\ &= 0.33\end{align}\)

Due to its high energy, such a particle might be involved in the unification of the strong and electroweak forces, but it would be unable to detect with present techniques due to its brief life in such an energy state.

Therefore, due to the Heisenberg uncertainty principle predicts such a particle's existence will be too brief, this theoretical energy threshold precludes the detection of such a particle.