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1: Some Cooking Recipes for Quantum Mechanics

  • Page ID
    16279
  • In this first lecture we will review the foundations of quantum mechanics at the level of a cooking recipe. This will enable us to use them later for the discussion of the atomic structure and interaction between atoms and light. This is the first lecture of the Advanced Atomic Physics course at Heidelberg University, as taught in the winter semester 2018/2019. It is intended for master students, which have a basic understanding of quantum mechanics and electromagnetism. In total, we will study multiple topics of modern atomic, molecular and optical physics over a total of roughly 25 lectures, where each lectures is approximately 90 minutes. The topics of the lectures will be discussed in more details in the associated tutorials.

    Principles of Quantum mechanics

    In Atomic, molecular, and optical (AMO) physics we will encounter the consequences of quantum mechanics all the time. So we will start out with a review of the basic ingredients to facilitate the later discussion of the experiments. Some good introductions can be found in (Basdevant 2002, Jean-Louis Basdevant 2006) (Quantum Mechanics, Vo...)(Quantum Mechanics, Vo...a). We will mostly follow the discussion of Ref. (Jean-Louis Basdevant 2006).

    Identify a Suitable Hilbert Space for the Problem in Question

    The first step is to identify the right Hilbert space for your problem. Mathematically, it will be a complex vector space with elements \(\left|\psi\right\rangle\)​ and a Hermitian scalar product

    \[ \langle\psi_{1}\psi_{2}\rangle=(\langle{\psi_{2}}|\psi_{1}\rangle)^{*} . \label{1}\]

    Some examples:

    1. Orbit in an atom, molecule etc. The structure of the Hilbert space will be discussed in more detail for the case of the hydrogen atom
    2. Occupation number of a photon mode. This Hilbert space is typically investigated in great detail in quantum optics.
    3. Position of an atom is of great importance for double slit experiments, the quantum simulation of condensed matter systems with atoms, or matterwave experiments.
    4. The arm of an interferometer is the canonical way of introducing phase dependent paths and detecting interference patterns.
    5. The spin degree of freedom of an atom like in the historical Stern-Gerlach experiment. The Hilbert space is now really simple as it reads:

    \[\left|\psi\right\rangle = a_{1}\left|\uparrow\right\rangle + a_{2}\left| \downarrow \right \rangle.\]

    with \(⟨ψ∣ψ⟩=1\).​​

    Observables and Measurements

    A given physical quantity \(A\)​ is associated with a Hermitian operator \(\hat{A}\)​ acting in the Hilbert space of the system. In a measurement, the possible outcomes are the eigenvalues \(a_{\alpha}\)​​ of the operator \(\hat{A}\)​:

    \[\hat{A}\left|\alpha\right\rangle=a_{\alpha}\left|\alpha\right\rangle. ​​\]

    The system will collapse to the corresponding eigenvector and the probability of finding the system in state \(\left|\alpha\right\rangle\)​ is

    \[ P(\left|\alpha\right\rangle) = ||\hat{P}_{\left|\alpha\right\rangle}\left|\psi\right\rangle||^{2} = \left\langle\psi\right | \hat{P}^{\dagger}_{\left|\alpha\right\rangle}\hat{P}_{\left|\alpha\right\rangle}\left | \psi\right\rangle,​​ \label{4}\]

    where \(\hat{P}_{\left|\alpha\right\rangle}=\left|\alpha\right\rangle\left\langle\alpha\right|​\).

    As for our previous examples, how would you measure them typically, i.e. what would be the operator ?

    1. The operator to be tested is the position operator of the electron bound to the nucleus. We access it indirectly through spectroscopy.
    2. For the occupation number we have nowadays number counting photodectors.
    3. The position of an atom might be detected through high-resolution CCD cameras.
    4. For the arm of the interferometer we will typically measure the output through photodetectors…
    5. For the measurement of the spin, we typically correlate the internal degree of freedom to the spatial degree of freedom. This is done by applying a magnetic field gradient acting on the magnetic moment \(\hat{\vec{\mu}}​\), which in turn is associated with the spin via \(\hat{\vec{\mu}}=g\mu_{B}\hat{\vec{s}}/\hbar​\), where \(g\)​ is the Landé-g​-factor and \(\mu_{B}\)​ is the Bohr magneton. The energy of the system is \(\hat{H}=-\hat{\vec{\mu}}\cdot\vec{B}​\).

    Complete Set of Commuting Observables

    A set of commuting operators \(\{\hat{A},\hat{B},\hat{C},\cdots,\hat{X}\}\) is considered a complete set if their common eigenbasis is unique. Thus, the measurement of all quantities \(\{A,B,\cdots,X\}\) ​will determine the system uniquely. The clean identification of such a Hilbert space can be quite challenging and a nice way of its measurment even more. Coming back to our previous examples:

    1. Performing the full spectroscopy of the atom. Even for the hydrogen atom we will see that the full answer can be rather involved…
    2. The occupation number is rather straight forward. However, we have to be careful that we really collect a substantial amount of the photons etc.
    3. Are we able to measure the full position information ? What is the resolution of the detector and the point-spread function ?
    4. Here it is again rather clean to put a very efficient detector at the output of the two arms …
    5. What are the components of the spin that we can access ? The \(z\)​ component does not commute with the other components, so what should we measure ?

    Time Evolution

    Being able to access the operator values and initialize the wavefunction in some way, we also want to have a prediction on its time-evolution. For most cases of this lecture we can simply describe the system by the non-relativistic Schrödinger Equation. It reads

    \[ i\hbar\partial_{t}\left|\psi(t)\right\rangle=\hat{H}(t)\left|\psi(t)\right\rangle. \label{5}\]

    In general, the Hamilton operator \(\hat{H}\) is time-dependent. For a time-independent Hamilton operator \(\hat{H}\), we can find eigenstates \(\left|\phi_{n}\right\rangle\)​ with corresponding eigenenergies \(E_{n}\)​:

    \[ \hat{H}\left|\phi_{n}\right\rangle=E_{n}\left|\phi_{n}\right\rangle. \label{6}\]

    The eigenstates \(\left|\phi_{n}\right\rangle\)​ in turn have a simple time evolution:

    \[ \left|\phi_{n}(t)\right\rangle=\left|\phi_{n}(0)\right\rangle\cdot\exp{-iE_{n}t/\hbar}.\label{7}\]

    If we know the initial state of a system

    \[ \left|\psi(0)\right\rangle=\sum_{n}\alpha_{n}\left|\phi_{n}\right\rangle,\label{8}\]

    where \(\alpha_{n}=\langle\phi_{n}|\psi(0)\rangle\)​, we will know the full dimension time evolution

    \[\left|\psi(t)\right\rangle=\sum_{n}\alpha_{n}\left|\phi_{n}\right\rangle\exp{-iE_{n}t/\hbar}.\;\,\text{(Schrödinger picture)} \label{9}\]

    Note

    Sometimes it is beneficial to work in the Heisenberg picture, which works with static ket vectors \(\left|\psi\right\rangle^{(H)}\)​ and incorporates the time evolution in the operators.1 In certain cases one would have to have access to relativistic dynamics, which are then described by the Dirac equation. However, we will only touch on this topic very briefly, as it directly leads us into the intriguing problems of quantum electrodynamics.

    Entangled States

    Consider a quantum system \(S\)​ formed by two subsystems \(S_{1}\)​​ and \(S_{2}\)​​. For each of them we can write:

    \[ \begin{align} \left|\psi_{1}\right\rangle & = \sum_{m}^{M}a_{m}\left|\alpha_{m}\right\rangle, \\ \left|\psi_{2}\right\rangle & =\sum_{n}^{N}b_{n}\left|\beta_{n}\right\rangle.​\label{10} \end{align}\]

    The question is now if it is always possible to write \(\left|\psi\right\rangle\)​ in the form

    \[ \begin{align} \left|\psi\right\rangle &= \left|\psi_{1}\right\rangle\otimes\left|\psi_{2}\right\rangle \\[5pt] &= \left(\sum_{m}^{M}a_{m}\left|\alpha_{m}\right\rangle\right) \otimes \left( \sum_{n}^{N} b_{n} \left| \beta_{n} \right | \rangle \right) \label{11} \\[5pt] &= \sum_{m}^{M} \sum_{n}^{N} a_{m} b_{n} \left| \alpha_{m} \right\rangle \otimes \left| \beta_{n} \right \rangle. \label{12} \end{align}\]

    From Equation \ref{12}, we see that the state \(\left|\psi\right\rangle\)​ is determined by \(M+N\)​ numbers. \(\left|\psi\right\rangle\)​ can also have the form of a superposition

    \[ \left|\psi\right\rangle=\sum_{m}^{M}\sum_{n}^{N}c_{mn}\left|\alpha_{m}\right\rangle\otimes\left|\beta_{n}\right\rangle.\label{13}\]

    Other than in Equation \ref{12}, \(\left|\psi\right\rangle\)​ is determined by \(M\times N\) numbers \(c_{mn}\)​ here. If we can not write \(\left|\psi\right\rangle\) as in Equation \ref{11}, we call the state entangled. Most states are entangled states. Equation \ref{11} thus only describes a small subset of all possible states.

    Example \(\PageIndex{1}\): Two Spins

    For a system of two spins we can construct the superposition state

    \[ \frac{1}{\sqrt{2}}(\left|\uparrow\right\rangle\otimes\left|\downarrow\right\rangle-\left|\downarrow\right\rangle\otimes\left|\uparrow\right\rangle).​\label{14} \nonumber\]

    The minus sign was arbitrarily chosen. Note that the first spin has a Hilbert space different from the Hilbert space of the second spin!

    Example \(\PageIndex{2}\): Electron in a hydrogen atom.

    We consider an electron orbiting around a proton. With the tensor product, we can connect the internal degree of freedom of the electron—given by its possible spin states—to its motional degree of freedom, which is characterized by the orbital quantum number \(l\)​ and the magnetic quantum number \(m_{l}\)​​:

    \[ \left\{ \begin{array}{c}\left|\uparrow\right\rangle\\ \left|\downarrow\right\rangle\end{array}\right\}\otimes\left\{ \begin{array}{c}\left|l=1,m_{l}=-1\right\rangle\\ \left|l=1,m_{l}=\enspace\;0\right\rangle\\ \left|l=1,m_{l}=+1\right\rangle\end{array}\right\} \label{15} \nonumber\]

    Unlike Example \(\PageIndex{1}\), it is obvious here that the Hilbert spaces differ from each other.

    The next lecture of this series can be found here: (Jendrzejewski).

    References

    • Dalibard Basdevant. Quantum Mechanics. Springer-Verlag, 2002. Link
    • Jean Dalibard Jean-Louis Basdevant. The Quantum Mechanics Solver. Springer-Verlag, 2006. Link
    • Quantum Mechanics, Volume 1. Link
    • Quantum Mechanics, Volume 2. Link
    • Fred Jendrzejewski, Selim Jochim, Matthias Weidem ller. Lecture 2 - A few more cooking recipes for quantum mechanics. Authorea Inc. Link