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## Quantum Mechanics I ### Problem Set 9 Due Thursday, November 16, 2023 at 11:59pm **Guidelines:** Please read the following guidelines carefully before starting the problem set. * **Typesetting:** You are required to submit your problem sets electronically through Canvas in PDF format. * *...

## Quantum Mechanics I ### Problem Set 9 Due Thursday, November 16, 2023 at 11:59pm **Guidelines:** Please read the following guidelines carefully before starting the problem set. * **Typesetting:** You are required to submit your problem sets electronically through Canvas in PDF format. * **Collaboration:** We encourage you to work together on the problem sets. However, each student must write up their solutions separately and independently. Please acknowledge at the beginning of your problem set all collaborators. * **Discussion:** Feel free to discuss the problem sets in the online forum. * **Grading:** Problem sets will be graded based on correctness, completeness, and clarity. Explain your reasoning clearly. ### Problem 1: Stark Effect (25 points) Consider the Hydrogen atom in a uniform external electric field along the $z$ direction, $\vec{E} = \varepsilon \hat{z}$. (a) Calculate the energy shift of the ground state to leading order in $\varepsilon$. (b) Calculate the energy shift of the $n = 2$ states to leading order in $\varepsilon$. Specify all the good quantum numbers you are using. (c) Calculate the induced electric dipole moment of the ground state to leading order in $\varepsilon$. ### Problem 2: Variational Method (25 points) Consider a particle of mass $m$ in a 1D potential $V(x) = \lambda |x|$, where $\lambda$ is a positive constant. (a) Use the variational method to estimate the ground state energy of this potential, using the trial wavefunction $\qquad \psi(x) = A e^{-bx^2}$, where $b$ is a variational parameter. (b) Compare your result with the exact ground state energy, which is given by $\qquad E_0 = \bigg( \frac{\lambda^2 \hbar^2}{2m} \bigg)^{1/3} a_1$, where $a_1 \approx 1.01879$ is the first zero of the Airy function $Ai(x)$. ### Problem 3: WKB Approximation (25 points) Consider a particle of mass $m$ in a potential $V(x) = \infty$ for $x < 0$, $V(x) = \frac{1}{2} m \omega^2 x^2$ for $x > 0$. Use the WKB approximation to estimate the allowed energy levels. ### Problem 4: Time-dependent Perturbation Theory (25 points) Consider a hydrogen atom initially in its ground state. At $t = 0$, it is placed in a time-dependent electric field along the $z$ direction, $\qquad \vec{E}(t) = \varepsilon e^{-t/\tau} \hat{z}$. Calculate the probability that the atom will be found in the $2p_z$ state after a long time $(t \rightarrow \infty)$.