Phy5646

Welcome to the Quantum Mechanics B PHY5646 Spring 2009

Schrodinger equation. The most fundamental equation of quantum mechanics which describes the rule according to which a state $|\Psi \rangle$ evolves in time.

This is the second semester of a two-semester graduate level sequence, the first being PHY5645 Quantum A. Its goal is to explain the concepts and mathematical methods of Quantum Mechanics, and to prepare a student to solve quantum mechanics problems arising in different physical applications. The emphasis of the courses is equally on conceptual grasp of the subject as well as on problem solving. This sequence of courses builds the foundation for more advanced courses and graduate research in experimental or theoretical physics.

The key component of the course is the collaborative student contribution to the course Wiki-textbook. Each team of students (see Phy5646 wiki-groups) is responsible for BOTH writing the assigned chapter AND editing chapters of others.

This course's website can be found here.

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Outline of the course:

Stationary state perturbation theory in Quantum Mechanics

Very often, quantum mechanical problems cannot be solved exactly. We have seen last semester that an approximate technique can be very useful since it gives us quantitative insight into a larger class of problems which do not admit exact solutions. The technique we used last semester was WKB, which holds in the asymptotic limit $\hbar \rightarrow 0$.

Perturbation theory is another very useful technique, which is also approximate, and attempts to find corrections to exact solutions in powers of the terms in the Hamiltonian which render the problem insoluble.

Typically, the (Hamiltonian) problem has the following structure

${\mathcal {H}}={\mathcal {H}}_{0}+{\mathcal {H}}'$

where ${\mathcal {H}}_{0}$ is exactly soluble and ${\mathcal {H}}'$ makes it insoluble.

Raleigh-Shrödinger Peturbation Theory

To do this we first consider an auxiliary problem, parameterized by $\lambda$:

${\mathcal {H}}={\mathcal {H}}_{0}+\lambda {\mathcal {H}}^{'}$

If we attempt to find eigenstates $|N(\lambda )\rangle$ and eigenvalues $E_{n}$ of the Hermitian operator ${\mathcal {H}}$, and assume that they can be expanded in a power series of $\lambda$:

$E(\lambda )=E_{0}+\lambda E_{1}+...+\lambda ^{n}E_{n}+...$

$|N(\lambda )\rangle =|\Psi _{n}^{(0)}\rangle +\lambda |\Psi _{n}^{(1)}\rangle +\lambda ^{2}|\Psi _{n}^{(2)}\rangle +...\lambda ^{j}|\Psi _{n}^{(j)}\rangle +...$

then they must obey the equation,

${\mathcal {H}}|N(\lambda )\rangle =E(\lambda )|N(\lambda )\rangle$.

Which, upon expansion, becomes:

$({\mathcal {H}}_{0}+\lambda {\mathcal {H}}')(\Sigma _{j=0}^{\infty }\lambda ^{j}|j\rangle )=(\Sigma _{l=0}^{\infty }\lambda ^{l}E_{l})(\Sigma _{j=0}^{\infty }\lambda ^{j}|j\rangle )$

We shall denote the unperturbed states as $|n\rangle$. We choose the normalization such that the unperturbed states are normalized, $\langle n|n\rangle =1$, and that the exact state satisfies $\langle n|N(\lambda )\rangle =1$. Note that in general $|N\rangle$ will not be normalized.

In order for this method to be useful, the perturbed energies must vary continuously with $\lambda$. Knowing this we can see several things about our, as yet undetermined peterbed energies and eigenstates. For one, as $\lambda \rightarrow 0,|N(\lambda )\rangle \rightarrow |n\rangle$, for some unperturbed state $|n\rangle$.

Perturbation correction eigenstates states are orthogonal unperturbed states,

Brillouin-Wigner Peturbation Theory

Interaction of radiation and matter

Quantization of electromagnetic radiation

Additional Reading

• Experimental observation of a Lamb-like shift in a solid state setup Science 322, 1357 (2008).

Classical view

Let's use transfer gauge (sometimes called Coulomb gauge):

$\varphi (\mathbf {r} ,t)=0$

$\nabla \mathbf {A} =0$

In this gauge the electromagnetic fields are given by:

$\mathbf {E} (\mathbf {r} ,t)=-{\frac {1}{c}}{\frac {\partial \mathbf {A} }{\partial t}}$

$\mathbf {B} (\mathbf {r} ,t)=-\nabla \times \mathbf {A}$

The energy in this radiation is

$\varepsilon ={\frac {1}{8\pi }}\int d^{3}\mathbf {r} (\mathbf {E} ^{2}+\mathbf {B} ^{2})$

The rate and direction of energy transfer are given by poynting vector

$\mathbf {P} ={\frac {c}{4\pi }}\mathbf {E} \times \mathbf {B}$

The radiation generated by classical current is

$\Box \mathbf {A} =-{\frac {4\pi }{c}}\mathbf {j}$

Where $\Box$ is the d'Alembert operator. Solutions in the region where $\mathbf {j} =0$ are given by

$\mathbf {A} (\mathbf {r} ,t)=\alpha \mathbf {\lambda } (\mathbf {k} )e^{i(\mathbf {k} \cdot \mathbf {r} -\omega t)}+\alpha ^{*}\mathbf {\lambda } ^{*}(\mathbf {k} )e^{-i(\mathbf {k} \cdot \mathbf {r} -\omega t)}$

where $\omega =c|\mathbf {k} |$ and $\mathbf {\lambda } (\mathbf {k} )\cdot \mathbf {k} =0$ in order to satisfy the transversality. We can choose $\mathbf {\lambda } \cdot \mathbf {\lambda } ^{*}=1$. Notice that in this writing $\mathbf {A}$ is a real vector.

Let's compute $\varepsilon$. For this

${\begin{aligned}\mathbf {E} (\mathbf {r} ,t)&=-{\frac {1}{c}}{\frac {\partial \mathbf {A} }{\partial t}}\\&=-{\frac {1}{c}}{\frac {\partial }{\partial }}\\&=\\&=\\\end{aligned}}$