Question

Equation (39.30) states that the energy of a system can have uncertainty. Does this mean that the principle of conservation of energy is no longer valid? Explain.

Short answer

Any system which is observed for very small time scale, the principle of conservation of energy is valid.

Step-by-step solution

State uncertainty principle for energy and time interval

A system that remains in a metastable state for a very long time (large ∆t) can have a very well-defined energy (small ∆E), but if it remains in a state for only a short time (small ∆t) the uncertainty in energy must be correspondingly greater (large ∆E).

$∆E∆t\ge \frac{ħ}{2\mathrm{\pi }}$

State principle of conservation of energy

The principle of energy conservation states that energy in any system can neither created nor destroyed.

Agreement of uncertainty principle for energy and time interval, and the principle of conservation of energy

In a system, it is possible to have energy coming out of nowhere but it can only last up to very small time interval that most of times it does not have any real effects on the system due to which principle of conservation of energy is not violated.

Hence, any system which is observed for very small time scale, the principle of conservation of energy is valid.