Expectation value of kinetic energy operator

In summary, the expectation value of the kinetic energy operator in the ground state ##\psi_0## is given by ##=\frac{\hbar w}{4}##, which coincides with classical mechanics where kinetic and potential energy are equal.
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The expectation value of the kinetic energy operator in the ground state ##\psi_0## is given by
$$<\psi_0|\frac{\hat{p^2}}{2m}|\psi_0>$$
$$=<\psi_0|\frac{1}{2m}\Big(-i\sqrt{\frac{\hbar mw}{2}}(\hat{a}-\hat{a^{\dagger}})\Big)^2|\psi_0>$$
$$=\frac{-\hbar w}{4}<\psi_0|\hat{a}^2+-\hat{a}\hat{a^{\dagger}}-\hat{a^{\dagger}}\hat{a}+\hat{a^{\dagger}}^2|\psi_0>$$
$$=\frac{-\hbar w}{4}<\psi_0|\Big(\hat{a}^2|\psi_0>-\hat{a}\hat{a^{\dagger}}|\psi_0>-\hat{a^{\dagger}}\hat{a}|\psi_0>+\hat{a^{\dagger}}^2|\psi_0>\Big)$$
$$=\frac{-\hbar w}{4}<\psi_0|\Big(|-\psi_0>+\sqrt{2}|\psi_2>\Big)$$
$$=\frac{-\hbar w}{4}<-\psi_0|\psi_0>+<\psi_0|\psi_2>$$

$$=\frac{\hbar w}{4}$$
 
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So we know for ground state energy as well as excited states energy kinetic energy and potential energy are half and half. That coincide with classical mechanics.
 
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1. What is the expectation value of the kinetic energy operator?

The expectation value of the kinetic energy operator is a measure of the average kinetic energy of a particle in a given quantum state. It is calculated by taking the inner product of the wave function with the kinetic energy operator and then integrating over all possible positions and momenta.

2. How is the expectation value of the kinetic energy operator related to uncertainty in position and momentum?

The expectation value of the kinetic energy operator is inversely proportional to the uncertainty in position and momentum. This means that as the uncertainty in position and momentum decreases, the expectation value of the kinetic energy operator increases.

3. Can the expectation value of the kinetic energy operator be negative?

No, the expectation value of the kinetic energy operator cannot be negative. This is because kinetic energy is always a positive quantity and the expectation value is calculated by taking the inner product of the wave function, which is a square of the amplitude, with the operator.

4. How does the expectation value of the kinetic energy operator change with the energy level of a particle?

The expectation value of the kinetic energy operator increases with the energy level of a particle. This is because as the energy level increases, the probability of the particle occupying higher energy states also increases, leading to a higher average kinetic energy.

5. How is the expectation value of the kinetic energy operator used in quantum mechanics?

The expectation value of the kinetic energy operator is a crucial concept in quantum mechanics as it is used to calculate various physical quantities such as the average velocity and the average force acting on a particle. It is also used to determine the energy spectrum of a system and to analyze the behavior of quantum systems.

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