# Energy-Time Uncertainty

In quantum mechanics, the uncertainty principle is undoubtedly one of the most famous concept. It is basically a mathematical inequality which puts a fundamental limit to the precision with which we can simultaneously measure certain pairs of physical properties of a particle. In our junior classes, we generally study about the uncertainty associated with the measurement of position and momentum of a particle. But, little do we know that such an uncertainty also exists for energy and time. In the 28th article of our Month of Equations series, we aim to discuss this Energy-Time Uncertainty in detail.

### Mathematical Expression:

• ?E - the uncertainty in the energy measurement
• ?t - the uncertainty in the lifetime measurement
• ? - h/2?, where h is the planck's constant

### Meaning of Energy-Time Uncertainty:

"In simple words, the Energy-Time uncertainty principle states that a quantum state that exists for only a short period of time cannot have a definite energy and vice versa"

### Mathematical formalism:

We know that the frequency of a state is inversely proportional to time. If the energy is in the form of EM waves, the limited time available restricts the accuracy with which we can determine the frequency ? of the waves.Moreover, the frequency directly has a link with the energy of the state, ie. ?E = h??. Eventually, using these relations, we can derive the above form of Energy-Time Uncertainty.

### How Energy-Time Uncertainty leads to Width of Spectral lines:

Apart from several other important phenomenons,The Energy-Time Uncertainty also results in giving a certain width to emitted spectral lines. To understand this, Let us consider the excited states of an atom. Each time an excited state decays, different energies are emitted. We characterize an emission line by a distribution of spectral frequencies of the emitted photons. All the spectral lines are characterized by their spectral widths. The average energy of the emitted photon corresponds to the theoretical energy of the excited state. It gives the spectral location of the peak of the emission line. It has been observed that the Short-lived states have broad spectral widths and long-lived states have narrow spectral widths.

### Author's message:

No doubt, The Energy-Time Uncertainty principle is more difficult to interpret than the position- momentum uncertainty principle. This is primarily because position and momentum and also energy are operators in quantum mechanics. Whereas, the status of time in quantum mechanics is a more difficult issue. But, an understanding of Energy-time uncertainty helps us to have useful insights into phenomenon related to tunneling, Unruh radiations, schwinger effect, ground state of EM field and a lot more. I hope that this article was helpful in giving you a brief overview of what energy-time uncertainty really stands for!

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