Virial theorem. Use Equation 3.71 to show that (d)/(d t)⟨x p⟩=2⟨T⟩-⟨x (d V)/(d x)⟩. where T is t

Virial theorem. Use Equation 3.71 to show that (d)/(d t)⟨x...

Virial theorem. Use Equation $3.71$ to show that $$ \frac{d}{d t}\langle x p\rangle=2\langle T\rangle-\left\langle x \frac{d V}{d x}\right\rangle . $$ where $T$ is the kinetic energy $(H=T+V) .$ In a stationary state the left side is zero (why?) so $$ 2\langle T\rangle=\left\langle x \frac{d V}{d x}\right\rangle $$ This is called the virial theorem. Use it to prove that $\langle T\rangle=\langle V\rangle$ for stationary states of the harmonic oscillator, and check that this is consistent with the results you got in Problems $2.11$ and $2.12$.

Virial theorem. Use Equation $3.71$ to show that
$$
\frac{d}{d t}\langle x p\rangle=2\langle T\rangle-\left\langle x \frac{d V}{d x}\right\rangle .
$$
where $T$ is the kinetic energy $(H=T+V) .$ In a stationary state the left side is zero (why?) so
$$
2\langle T\rangle=\left\langle x \frac{d V}{d x}\right\rangle
$$
This is called the virial theorem. Use it to prove that $\langle T\rangle=\langle V\rangle$ for stationary states of the harmonic oscillator, and check that this is consistent with the results you got in Problems $2.11$ and $2.12$.

Key Concepts

Virial Theorem

The virial theorem is a fundamental result in both classical and quantum mechanics that relates the average kinetic energy of a bound system to its potential energy. In quantum mechanics, this theorem connects expectation values of kinetic and potential energy operators, providing insights into the balance of forces in stationary or equilibrium states.

Stationary States

Stationary states are quantum states that are eigenstates of the Hamiltonian operator, meaning they have a definite energy and their probability distributions do not change over time. For such states, any time derivative of an expectation value of an operator that is not explicitly time-dependent will vanish, a fact that is crucial when applying the virial theorem in these systems.

Expectation Values and Operator Equations of Motion

Expectation values in quantum mechanics represent the average outcome of measurements over many experiments and are computed using the state of the system. The time evolution of these expectation values can be determined by the Heisenberg equation of motion, which involves commutators of the observable with the Hamiltonian, allowing one to relate dynamics with physical quantities like kinetic and potential energy.

Kinetic and Potential Energy Relation

In many quantum systems, particularly those in equilibrium or stationary states, there exists a balance between kinetic and potential energy as stipulated by the virial theorem. This balance is expressed by formulas that equate or relate the expectation values of these energy components, revealing underlying symmetries in the potential energy function.

Harmonic Oscillator

The harmonic oscillator is a paradigmatic quantum system where the potential energy is quadratic in displacement. Its simplicity allows for exact solutions, and the virial theorem applies to show that, in stationary states, the expectation values of the kinetic and potential energies are equal. This equality is one of the defining characteristics of the harmonic oscillator and serves as a consistency check for more general results derived using the virial theorem.

Transcript

00:04 This is the problem number 31 and this theorem is based on the virile theorem and the problem is said that you the equation 3 .7 to show that that we have to use the equation 3 .7 that is very real theorem using equation 3 .7 we have to show that d by the derivative of this xp is 2 of t minus x d v y d x these are the expectation value where t is the kinetic energy and h is equal to t plus b and then you have to show that in a stationary state the left side this is 0 and you have to explain why this is 0 and so that 2 t is equal to x j b d x and we have to use some equation in this problem we have it is mentioned in this question that we have to use the equation 3 .71 we have equation 3 .71 which is the derivative of d y d t xp is equal to i y h bar expectation of the commutation relation of h and xp and now we should know the value of this h xp the from equation 3 .64 the value of this xp that is commutation relation of the h with xp is h xp plus x and hp here h and p here h and p are the operator and from equation 3 .14 we have the computational relation of the h with p is i h bar and d by d x so d y d t x p is this equation is from this equation let's write this is one this is two this is three now equation one becomes to this i by h bar and this h xp is this p plus xp now also should the value of this that is i by h bar x x x x is okay 3 .1 we have another one relation that is h x is equal to minus i s par p by n this is 3 .14 and this is 3 .17 d this equation from 3 .17 this equation for 3 .14 let's say this is 3 and this is 4.

05:14 Now so it will do all values here at x that's minus i s bar p by n minus i s bar p by n and this is p plus x and hp is i h bar d b by d x and here this is commutation that is expectation value of this this is like this we have to write this one now write separately here that is i by h bar and this value will be minus i h bar expectation that is b square by m plus i by h bar and and this is i h bar and here that is x d b by d x this i square minus 1 and minus 1 that is plus that is s bar if i multiply by 2 here and 2 divide that it is p squared by 2m and this this is h square by 2m and this is minus x dby by d x and this is nothing this is the kinetic energy that is expectation of the kinetic energy minus x dv by d x and this is derivative of x and now we have to show that in a stationary state the left side is zero in this equation and this equation is 5.

07:57 For stationary state, the expectation value are, does not depend on the, that is time independent...

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