Short Answer

Expectation value is the average value of a physical quantity in quantum mechanics. It is the value we expect to obtain if we measure the same quantity many times on identical quantum systems. Instead of giving a single definite value, quantum mechanics provides this statistical average.

The expectation value is calculated using the wave function of the system. It helps describe measurable quantities such as position, momentum, energy, and angular momentum. Expectation values play an important role because many quantum properties cannot be known exactly, only in terms of averages.

Detailed Explanation :

Expectation value

Expectation value is a fundamental concept in quantum mechanics that describes the average result of measuring a physical property of a quantum system. In classical physics, physical quantities like position, momentum, and energy have definite values. However, in quantum mechanics, particles do not have definite values until they are measured. Instead, these quantities are represented by probability distributions derived from the wave function. The expectation value provides the most likely average value of a quantity if the measurement is repeated many times on identical systems.

The expectation value connects the mathematical structure of quantum mechanics with physical observables. It tells us what we can expect to measure, even though individual measurements may give different results. This is because quantum states are probabilistic in nature, not deterministic like classical states.

Meaning of expectation value

Expectation value represents the statistical mean of a physical quantity. If we could perform the same measurement repeatedly on many identical particles prepared in the same state, the outcomes would vary but cluster around a central value. That central value is the expectation value.

For example, if we measure the position of an electron many times, the exact value will differ each time because of uncertainty. But the expectation value of position tells us the average position predicted by the wave function.

Mathematical expression

In quantum mechanics, physical quantities are represented by operators. If  is an operator representing a quantity A, and ψ is the wave function of the system, then the expectation value of A is given by:

Here,

•  = wave function
•  = complex conjugate of the wave function
•  = operator corresponding to the physical quantity
• dv = volume element

This formula shows how quantum mechanics uses probability densities to calculate measurable averages.

Expectation value and wave function

The wave function contains all information about a system. The square of the wave function (ψ²) gives the probability density. The expectation value uses this probability density to calculate the average value of any observable.

Examples:

• Expectation value of position uses the position operator .
• Expectation value of momentum uses the momentum operator .
• Expectation value of energy uses the Hamiltonian operator.

Thus, expectation values allow us to extract physical information from wave functions.

Expectation value in position

For position, the expectation value is:

This gives the average position of the particle. It does not mean the particle stays at this value; rather, it is the statistical centre of the probability distribution.

Expectation value in momentum

For momentum, the expectation value is:

This helps describe the most likely momentum of the particle.

Expectation value in energy

Energy expectation values show the average energy of a system. For bound states, the expectation value often matches quantized energy levels obtained from solving Schrödinger’s equation.

Importance of expectation values

Expectation values are essential in quantum mechanics because:

1. They connect theory with experiments
   Quantum mechanics is probabilistic. Expectation values give the measurable averages that appear in experiments.
2. They describe physical properties
   Quantities like position, momentum, and energy are understood in terms of their expectation values.
3. They help evaluate system behaviour
   Expectation values change with time, helping us study how a quantum system evolves.
4. Used in Ehrenfest’s theorem
   This theorem links quantum expectation values to classical laws of motion, showing that classical physics emerges from quantum physics.
5. Useful in atomic and molecular physics
   Expectation values help calculate average distances, energies, and forces within atoms and molecules.

Expectation value vs most probable value

Expectation value is the average value, not the most likely value. For example, the density of water in a glass has an expectation value based on distribution, but the most probable value is the highest point of the distribution. In quantum mechanics, these two concepts can be different.

Expectation values and measurement

Individual quantum measurements give random outcomes because of probability. But the expectation value matches the average of many measurements. This is why expectation values are more meaningful in quantum theory than a single measurement.

Applications of expectation values

Expectation values are used in:

• atomic physics (electron energy levels)
• molecular physics (bond lengths and energies)
• solid-state physics (electron transport)
• quantum chemistry (electron density)
• quantum computing (qubit behaviour)
• statistical mechanics

They form the bridge between abstract quantum theory and real experimental results.

Conclusion

Expectation value is the average value of a physical quantity in a quantum system. It is calculated using the wave function and the operator that represents the quantity. Expectation values help interpret measurable properties such as position, momentum, and energy in probabilistic systems. They connect quantum mechanics with real-world observations and form a key part of understanding microscopic behaviour.