1. For the ground state of the one-dimensional harmonic oscillator, find the average value of the kinetic and potential energies. Check that <T>=<V> in this case. Find the most probable position of the particle for said wave function and verify that the uncertainty principle is satisfied. Data:$\int_{0}^{\infty}x^2 e^{-\alpha x^2}dx=\pi^{1/2}/4\alpha^{3/2}$

2. Using the Rodrigues formula, generate the polynomials with $v=0,1,2,3....$

3. The three-dimensional harmonic oscillator has the potential energy function $V(x,yz)=k_x \frac{x^2}{2} + k_y \frac{y^2}{2} + k_z \frac{z^2} {2}$, where $k_x$, $k_y$, and $k_z$ are three force constants. Using the method of separation of variables, find the eigenfunctions and the eigenvalues of the energy of said oscillator.

4. Find the eigenfunctions and eigenvalues of the Hamiltonian $H$ for a one-dimensional system with $V(x)=\infty$ for $x<0$ and $V(x)=\frac{1}{2}kx^{ 2}$ for $x\geq 0$.

5. Show explicitly that Hermite polynomials with $v=2,3,4$ satisfy the recurrence relation: $\xi H_{v-1} = (v-1)H_{v-2} + \frac{1} {2}H_{v}$.

6. Prove the following recurrence relation between Hermite polynomials: $H_v = 2\xi H_{v-1}-H'_{v-1}$

7. The inversion operator $\hat{i}$ converts each point in space into its opposite: $\hat{i}f(x,y,z)=f(-x,-y,-z)$, where $f$ is any function. Determines the eigenvalues of this operator and sets the characteristics of its eigenfunctions. In the case of a cubic box or the 3D isotropic harmonic oscillator, verify that the Hamiltonian operator commutes with the inversion operator, and that the standing and separable wave functions of both systems are also eigenfunctions of inversion.

8. Consider an isotropic two-dimensional harmonic oscillator whose potential is $V(x,y)=\frac{1}{2}k\left(x^2+y^2\right)$. Write the Schrödinger equation of this system, and the general expressions of the state functions and stationary energies that are obtained by solving it using the separation of variables technique. The solution of the one-dimensional harmonic oscillator is $\Psi_{v_{x}}=N_{v_{x}}H_{v_{x}}e^{-\xi^2_x/2}$, $E_{v_{x }}=\left(v_{x}+\frac{1}{2}\right)h\nu_{x}$ The virial theorem for this system tells us that $\left\langle \hat{T }\right\rangle=\left\langle \hat{V}\right\rangle$. Use this result to calculate $\left\langle x^2+y^2\right\rangle$. For the ground state of the two-dimensional harmonic oscillator, check that the expressions for $\left\langle x^2\right\rangle$ and $\left\langle y^2\right\rangle$ are equivalent. Now calculate $\left\langle x^2\right\rangle$ also using the result from the previous section.

9. Using parity criteria, compute $\left\langle xy\right\rangle$ for any state of the two-dimensional harmonic oscillator. Consider the wave function $\Psi=\Psi_{1}-2\Psi_{2}+\Psi_{ 3}$, where $\Psi_{1}$, $\Psi_{2}$,$\Psi_{3}$ are normalized eigenfunctions of the Hamiltonian of the one-dimensional harmonic oscillator with v=1,2,3 respectively. a) Normalize $\Psi$. b) Find the values that can be obtained by measuring the energy of a one-dimensional harmonic oscillator that is in the state described by $\Psi$. c) Also find their corresponding probabilities. d) A particle is subjected to the potential $V(x)=ax^2-bx^3$. Can the proper functions of the Hamiltonian of this system be eigenfunctions of the parity operator, $\hat{\Pi}$? Justify the answer.

10. For the one-dimensional harmonic oscillator the solutions of the Schödinger equation are: $E_{v}=\hbar\omega\left(v+1/2\right)$ and $\Psi_{v}(x)=N_{v }H_{v}e^{-?{x}^2/2}$. The circular harmonic oscillator is a one-particle system that has an energy function given by $V(x,y)=\frac{1}{2}K\left(x^2+y^2\right)$. For this system: a) Derive the general expression of energy. b) Calculate the degeneracy of the energy levels Considering that $H_{0}=1$, find the expected value of $x^2y^3$ in the ground state of the circular harmonic oscillator.

11. The recurrence rule for the coefficients of Hermite polynomials is: $$a_{i+2}=\frac{2i-\left(\frac{\alpha}{\beta}-1\right)}{\left (i+2\right)\left(i+1\right)}a_{i}$$ Where $\alpha=\frac{2mE}{\hbar^2}$, $\beta=\left(\frac {mk}{\hbar^2}\right)^{1/2}$. If $a_{5}$ is the nonzero coefficient of the largest exponent, write an expression for E in terms of $\omega$ and $\hbar$. Write the time-independent Schödinger equation for a two-dimensional harmonic oscillator with $k_{x}=k_{y}$. Write the expression for $\Psi(x,y)$ and E using the one-dimensional harmonic oscillator solution.