- 5 Postulates of Quantum Mechanics
- 5.1 ๐ฅ Learning Objectives
- 5.2 Postulates of Quantum Mechanics
- 5.2.1
**Completeness**: All Observable Properties of a System can be Inferred from its Wavefunction - 5.2.2
**Schrodinger Postulate**: The wavefunction is found by solving the Schrรถdinger equation. - 5.2.3
**Born Postulate**: $\\left|\\Psi(x,t) \\right|^2$ is the probability of observing the system at position $x$ at time $t$. - 5.2.4
**Correspondence Principle**: Every observable operator corresponds to a linear Hermitian operator.

- 5.2.1
- 5.3 Completeness of $\\Psi$ as a descriptor of a quantum system
- 5.4 Schrรถdinger Postulate
- 5.5 Born Postulate
- 5.6 Correspondence Principle and Hermitian Operators
- 5.7 Bra- Ket- Notation and the Analogy to Linear Algebra
- 5.8 Application: Expansion in a Basis Set
- 5.9 Application: Heisenberg Uncertainty Principle
- 5.10 Application: Variational Principle
- 5.11 Summary
- 5.12 ๐ช Self-Reflection
- 5.13 ๐ค Thought-Provoking Questions
- 5.14 ๐ Recapitulation
- 5.15 ๐ฎ Next Up...
- 5.16 ๐ References

= \\sum_{k=0}^{k=\\infty} |c_k|^2 q_k = \\int \\left( \\Psi(x) \\right)^* \\hat{Q} \\Psi(x) dx\n", "$$\n", "\n", "This formula for the expectation value only holds for normalized wavefunctions. When the wavefunction is not normalized, instead one must use:\n", "$$\n", "\\left= \\frac{\\sum_{k=0}^{k=\\infty} |c_k|^2 q_k}{\\sum_{k=0}^{k=\\infty} |c_k|^2} = \\frac{\\int \\left( \\Psi(x) \\right)^* \\hat{Q} \\Psi(x) dx}{\\int \\left( \\Psi(x) \\right)^* \\Psi(x) dx}\n", "$$\n" ] }, { "cell_type": "markdown", "id": "egyptian-water", "metadata": {}, "source": [ "#### 📝 Exercise: Show that the equality in the last equation is true\n", "> Hint: expand the wavefunction in the eigenbasis, use the eigenvalue relation and the orthonormality of the eigenvectors." ] }, { "cell_type": "markdown", "id": "bigger-thesis", "metadata": {}, "source": [ "Immediately after performing a measurement of $Q$ for the system defined by $\\Psi(x)$, one knows definitively that the state of the system is described by $\\Psi(x) = \\psi_k(x)$, with eigenvalue $q_k$. This seems weird, as the wavefunction seems to have changed abruptly from $\\Psi(x)$ to $\\psi_k(x)$ *because* of the measurement. This would somehow imply that if the wavefunction for Schrödinger's cat were:\n", "$$\n", "\\Psi_{cat} = \\tfrac{1}{\\sqrt{2}}|\\text{alive} \\rangle + \\tfrac{1}{\\sqrt{2}}|\\text{dead} \\rangle\n", "$$\n", "and you opened the box and observed that the cat was dead (so after you open the box, $\\Psi_{cat} = |\\text{dead} \\rangle$), then *you* killed Schrödinger's cat. To mildly exaggerate, some physicists would have you believe that every dead animal was slaughtered by the person who first observes its corpse. (To diminish culpability, it must be said that it the cat in this example was only technically half-dead, so the observer was a halfway-cat-assassin.) Most modern [interpretations of quantum mechanics](https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics) tend to [deny such culpability](https://plato.stanford.edu/entries/qt-issues/). The physicists alibi is to assert that assert that while the system was described, mathematically, by $\\Psi(x)$ prior to the measurement, this does not that the system existed in the state $\\Psi(x)$. Similarly, after the measurement the system is in a state mathematically described by $\\psi_k(x)$. While it would be weird for observing a system to be able to change its state, it is not weird for an observation to change our mathematical description of a system. For example, before you observe [Lake Wobegon](https://en.wikipedia.org/wiki/Lake_Wobegon), it is reasonable to assume that all the women are strong, all the men are good-looking, and all the children are above average. But were you to visit Lake Wobegon, then based on your observation you might have to change your model. \n", "\n", "That said, you may find the aforementioned \"Copenhagen interpretation\" convenient. Before my mother visits my home, I always clean it thoroughly. Nonetheless, my thorough cleaning is not up to my mother's standards, and she's always scandalized to find dust-bunnies under the sofa. (Who moves the sofa to vacuum under it, just to put the sofa back the same place and obscure the now-clean carpet?) I always tell my mom that the dust-bunnies were not there until she observed them. Unfortunately, my mom taught quantum mechanics herself, and she tells me that the wavefunction was:\n", "$$\n", "\\Psi_{dust-bunnies} = \\sqrt{.0001}|\\text{clean} \\rangle + \\sqrt{.9999}|\\text{dirty} \\rangle\n", "$$\n", "I guess my home is only .01% clean (up to my mother's standards, at least). \n", "\n", "The sudden change in the wavefunction upon observation is called wavefunction collapse. While the reality of wavefunction collapse in a physical sense is irresolvable, it is the mathematical description of what happens in a quantum system, and gives rise to strange quantum effects. If you find such things interesting, you may be interested to know that the aphorism that \"a watched pot never boils\" is [justifiable](https://en.wikipedia.org/wiki/Quantum_Zeno_effect), quantum-mechanically." ] }, { "cell_type": "markdown", "id": "incredible-translator", "metadata": {}, "source": [ "### The Born Postulate, revisited\n", "While the Born postulate is usually presented separately, it is in fact a corollary of the fact that physical observables are represented by Hermitian operators. The Hermitian operator that represents a particle at position $x_0$ is $\\delta(x-x_0)$, where the [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function) is effectively defined by its so-called *sifting property*.\n", "> **Sifting property of the Dirac Delta function:** Let $f(x)$ be a bounded function. Then \n", "$$\n", "\\int_{-\\infty}^{+\\infty} f(x) \\delta(x-x_0) dx = f(x_0)\n", "$$\n", "\n", "Now according to the Hermitian postulate, the probability of observing the particle at position $x_0$ is given by \n", "$$\n", "\\int_{-\\infty}^{+\\infty} \\left(\\Psi(x)\\right)^* \\delta(x-x_0) \\Psi(x) dx = \\left|\\Psi(x_0)\\right|^2\n", "$$\n", "which is the Born postulate. \n", "\n", "> **Note**: The Kronecker delta function has a sifting property similar to the Dirac delta function,\n", "$$\n", "\\sum_j f_j \\delta_{jk} = f_k\n", "$$\n", "This can be deduced directly from the definition, since $\\delta_{jk}$ is one when $j=k$, and zero otherwise." ] }, { "cell_type": "markdown", "id": "pressing-reset", "metadata": {}, "source": [ "## Bra- Ket- Notation and the Analogy to Linear Algebra\n", "At this stage, your hand may be starting to hurt from writing integrals and wavefunctions. This is the motivation for [Dirac bra-ket notation](https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation). A rather thorough explanation of this notation is available in my [pdf notes](https://github.com/PaulWAyers/IntroQChem/blob/main/documents/LinAlgAnalogy.pdf?raw=true), so I am only presenting the elements here. The basic idea is that a wavefunction is written as a *ket*, \n", "$$\n", "| \\Psi \\rangle = \\Psi(x)\n", "$$\n", "and its complex conjugate as a *bra*,\n", "$$\n", "\\langle \\Phi | = \\left(\\Phi(x)\\right)^*\n", "$$\n", "The overlap between two wavefunctions, which as we shall see is very important when expanding a wavefunction as a linear combination of basis functions, is therefore:\n", "$$\n", "\\langle \\Phi | \\Psi \\rangle = \\int \\left(\\Phi(x)\\right)^* \\Psi(x) dx\n", "$$\n", "Because a Hermitian operator could act either forward (towards the ket)\n", "$$\n", "\\langle \\Phi |\\hat{Q} \\Psi \\rangle = \\int \\left(\\Phi(x)\\right)^* \\hat{Q} \\Psi(x) dx\n", "$$\n", "or backwards (towards the bra)\n", "$$\n", "\\langle\\hat{Q} \\Phi | \\Psi \\rangle = \\int \\left(\\hat{Q} \\Phi(x)\\right)^* \\Psi(x) dx\n", "$$\n", "we often use a notation that makes it clear that the operator can act in either direction, \n", "$$\n", "\\langle \\Phi | \\hat{Q} |\\Psi \\rangle = \\int \\left(\\Phi(x)\\right)^* \\hat{Q} \\Psi(x) dx\n", "$$" ] }, { "cell_type": "markdown", "id": "serious-glenn", "metadata": {}, "source": [ "## Application: Expansion in a Basis Set\n", "A set of functions, $\\{\\phi_k(x) \\}$ is said to be a complete basis set if any wavefunction can be written exactly as a linear combination of these basis functions,\n", "$$\n", "\\Psi(x) = \\sum_{k=0}^{\\infty} c_k \\phi_k(x)\n", "$$\n", "It is convenient and it is always possible, but it is not required, to choose the basis functions so that they are orthonormal, \n", "$$\n", "\\int \\left(\\phi_j(x) \\right)^* \\phi_k(x) dx = \\delta_{jk}\n", "$$\n", "where we have used the Kronecker-delta notation we introduced above.\n", "\n", "In bra-ket notation, the preceding equations are, respectively,\n", "$$\n", "|{\\Psi}\\rangle = \\sum_{k=0}^{\\infty} c_k |\\phi_k\\rangle\n", "$$\n", "and\n", "$$\n", "\\langle \\phi_j | \\phi_k \\rangle = \\delta_{jk}\n", "$$\n", "\n", "To obtain an equation for the expansion coefficient, multiply both sides of the first equation by $(\\phi_j(x))^*$ and integrate over $x$. This gives:\n", "$$\n", "\\int \\left( \\phi_j(x) \\right)^* \\Psi(x) dx = \\int \\left( \\phi_j(x) \\right)^* \n", " \\sum_{k=0}^{\\infty} c_k \\phi_k(x) dx\n", "$$\n", "Because the integral of a sum is the sum of the integrals, and because the integral of a constant is a constant times the integral, this simplifies to:\n", "$$\n", "\\begin{align}\n", "\\int \\left( \\phi_j(x) \\right)^* \\Psi(x) dx &=\\sum_{k=0}^{\\infty} c_k \\left[ \\int \\left( \\phi_j(x) \\right)^* \n", " \\phi_k(x) dx \\right] \\\\\n", "&=\\sum_{k=0}^{\\infty} c_k \\delta_{jk}\n", "\\end{align}\n", "$$\n", "Then from the sifting property of the Dirac delta function,\n", "$$\n", "c_j = \\int \\left( \\phi_j(x) \\right)^* \\Psi(x) dx \n", "$$\n", "*This is the equation for the expansion coefficient of a wavefunction in an orthonormal basis.*" ] }, { "cell_type": "markdown", "id": "certain-hazard", "metadata": {}, "source": [ "## Application: Heisenberg Uncertainty Principle\n", "We have already alluded to the [Heisenberg Uncertainty Principle](https://en.wikipedia.org/wiki/Uncertainty_principle), which states that some quantum-mechanical properties cannot be observed simultaneously. To provide a mathematical description of the Heisenberg Uncertainty Principle, we need to define what it means for operators to commute and anticommute.\n", "> Two operators, $\\hat{A}$ and $\\hat{B}$, are said to *commute* if for any wavefunction $\\Psi(x)$, \n", "$$0 = \\left(\\hat{A} \\hat{B} - \\hat{B}\\hat{A}\\right) \\Psi(x) = \\left[\\hat{A}, \\hat{B} \\right] \\Psi(x)\n", "$$\n", "Similarly, two operators are said to *anticommute* if\n", "$$0 = \\left(\\hat{A} \\hat{B} + \\hat{B}\\hat{A}\\right) \\Psi(x) = \\left\\{\\hat{A}, \\hat{B} \\right\\} \\Psi(x)\n", "$$\n", "In the right-most equality, we have introduce the standard notation for commuting and anticommuting operators.\n", "\n", "In classical mechanics, observables are simply functions of the momenta and positions of the system's particles, and the since the momenta and positions commute, observables commute. However, in quantum mechanics, the momentum operator, $\\hat{p} = -i \\hbar \\tfrac{d}{dx}$ is a differential operator, and does not commute with the position operator $x$. Because of this, measuring a particle's position first, then its momentum is different from measuring its momentum first, then its position. Conceptually, then, it is not unreasonable that if you try to measure the position and the momentum simultaneously, the system is \"confused\" about how it should behave (as if momentum were measured first? or as if position were measured first?) and the answer is uncertain. \n", "\n", "> If two operators, $\\hat{A}$ and $\\hat{B}$, commute, $\\left[\\hat{A}, \\hat{B}, \\right] = 0$. then one can simultaneously measure the corresponding properties $A$ and $B$. $A$ and $B$ are said to be *simultaneous observables*. \n", "\n", "> If two operators, $\\hat{A}$ and $\\hat{B}$, do not commute, $\\left[\\hat{A}, \\hat{B}, \\right] \\ne 0$. then one cannot simultaneously measure the corresponding properties $A$ and $B$. A Heisenberg Uncertainty Relation then holds: \n", "$$\n", "\\sigma_A^2 \\sigma_B^2 \\ge \\tfrac{1}{4} \\left| \\langle \\Psi |[\\hat{A},\\hat{B}]| \\Psi \\rangle \\right|^2\n", "$$\n", "where the variance in the expectation value of the operator is defined as:\n", "$$\n", "\\sigma_A^2 = \\langle \\Psi |\\hat{A}^2| \\Psi \\rangle - \\langle \\Psi |\\hat{A}| \\Psi \\rangle^2\n", "$$\n", "\n", "This expression can be made somewhat more precise by using the [Schrödinger Uncertainty Principle](https://en.wikipedia.org/wiki/Uncertainty_principle#Robertson%E2%80%93Schr%C3%B6dinger_uncertainty_relations)\n", "$$\n", "\\sigma_A^2 \\sigma_B^2 \\ge \\left| \\tfrac{1}{2}\\langle \\Psi |\\{\\hat{A},\\hat{B}\\}| \\Psi \\rangle \n", "- \\langle \\Psi |\\hat{A}| \\Psi \\rangle \\langle \\Psi |\\hat{B}| \\Psi \\rangle \\right|^2 + \\tfrac{1}{4} \\left|\\langle \\Psi |[\\hat{A},\\hat{B}]| \\Psi \\rangle \\right|^2\n", "$$\n", "\n", "A detailed derivation of these uncertainty principles is provided as a [pdf](https://github.com/PaulWAyers/IntroQChem/blob/main/documents/Heisenberg.pdf?raw=true)." ] }, { "cell_type": "markdown", "id": "accomplished-humidity", "metadata": {}, "source": [ "## Application: Variational Principle\n", "> [**Quantum-Mechanical Variational Principle**](https://en.wikipedia.org/wiki/Variational_method_(quantum_mechanics)): Given a wavefunction $\\Psi(x)$ and a bounded Hermitian operator $\\hat{Q}$, the expectation value of $Q$ is no less than the lowest eigenvalue of $\\hat{Q}$ and no greater than the largest eigenvalue.\n", "\n", "To understand where this principle comes from, let's focus on the lowest eigenvalue of $\\hat{Q}$. We denote the eigenvalues and eigenvectors of $\\hat{Q}$ as\n", "$$\n", "\\hat{Q} | \\psi_k \\rangle = q_k | \\psi_k \\rangle\n", "$$\n", "and choose to list the eigenvalues are listed in increasing order, $q_0 < q_1 < \\cdots $. Expand the wavefunction in the eigenvectors of $\\hat{Q}$,\n", "$$\n", "| Psi \\rangle = \\sum_{k=0}^{\\infty} c_k | \\psi_k \\rangle\n", "$$\n", "The expectation value of $\\Psi$, which we assume to be normalized, is:\n", "\\begin{align}\n", "\\langle \\Psi |\\hat{Q}|\\Psi \\rangle &= \\langle \\sum_{j=0}^{\\infty} c_j \\psi_j |\\hat{Q}|\\sum_{k=0}^{\\infty} c_k \\psi_k \\rangle \\\\\n", "&= \\sum_{j=0}^{\\infty} \\sum_{k=0}^{\\infty} c_j c_k \\langle \\psi_j |\\hat{Q}|\\psi_k \\rangle \\\\\n", "&= \\sum_{j=0}^{\\infty} \\sum_{k=0}^{\\infty} c_j c_k \\langle \\psi_j |\\hat{Q}\\psi_k \\rangle \\\\\n", "&= \\sum_{j=0}^{\\infty} \\sum_{k=0}^{\\infty} c_j c_k \\langle \\psi_j |q_k\\psi_k \\rangle \\\\\n", "&= \\sum_{j=0}^{\\infty} \\sum_{k=0}^{\\infty} c_j c_k q_k \\langle \\psi_j |\\psi_k \\rangle \\\\\n", "&= \\sum_{j=0}^{\\infty} \\sum_{k=0}^{\\infty} c_j c_k q_k \\delta_{jk} \\\\\n", "&= \\sum_{j=0}^{\\infty} |c_j|^2 q_j \n", "\\end{align}\n", "In the last line we used the sifting property of the Kronecker delta function. To complete the demonstration, recall that because the eigenvalues are listed in nondecreasing order, $q_j - q_0 \\ge 0$ for all $j$. So:\n", "\\begin{align}\n", "\\langle \\Psi |\\hat{Q}|\\Psi \\rangle &= \\sum_{j=0}^{\\infty} |c_j|^2 q_j \\\\\n", " &= \\sum_{j=0}^{\\infty} |c_j|^2 \\left(q_0 + (q_j - q_0)\\right) \\\\\n", " &= \\sum_{j=0}^{\\infty} |c_j|^2 q_0 + \\sum_{j=0}^{\\infty} |c_j|^2 (q_j - q_0) \\\\\n", " &= q_0 \\sum_{j=0}^{\\infty} |c_j|^2 + \\sum_{j=0}^{\\infty} |c_j|^2 (q_j - q_0) \\\\\n", " &= q_0 + \\sum_{j=0}^{\\infty} |c_j|^2 (q_j - q_0) \\\\\n", " &\\ge q_0 \n", "\\end{align}\n", "In the next to last line the normalization of the wavefunction, which implies that $\\sum |c_j|^2 = 1$, was used. In the last line a nonnegative term was eliminated from the expression, resulting in the desired inequality.\n", "\n", "This principle is especially useful for the energy, because it means that an approximate wavefunction will always have an energy greater than or equal to the true ground state energy. Therefore, if one is given two wavefunctions $\\Psi(x)$ and $\\Phi(x)$, the \"better\" ground-state wavefunction would be the one that has the lower energy. Similarly, given a normalized wavefunction that depends on a parameter, $\\kappa$, the \"best\" wavefunction and the \"best\" energy can be obtained by minimizing the expectation value of the energy with respect to $\\kappa$:\n", "$$\n", "\\min_\\kappa \\langle \\Psi(\\kappa) |\\hat{H} | \\Psi(\\kappa) \\rangle\n", "$$\n", "\n", "The variational principle is one of the most important ways we approximate the energy and wavefunction of quantum systems." ] }, { "cell_type": "markdown", "id": "anonymous-present", "metadata": {}, "source": [ "### 📝 Exercise: Show why the expectation value of $Q$ is always a lower bound on the largest eigenvalue of $\\hat{Q}$." ] }, { "cell_type": "markdown", "id": "joined-palestinian", "metadata": {}, "source": [ "## Summary\n", "Based on the preceding, the key postulates of quantum mechanics are \n", "- the Schrödinger equation determines the wavefunction\n", "- the wavefunction determines all observable properties of a quantum system\n", "- observable properties of a quantum system, and their observable values, correspond to Hermitian operators\n", "You should be able to apply, and expand upon, these principles at length." ] }, { "cell_type": "markdown", "id": "posted-aircraft", "metadata": {}, "source": [ "## 🪞 Self-Reflection\n", "- Try to come up with your own version of a Schrödinger's cat paradox. Have [fun with it](https://www.straightdope.com/21341296/the-story-of-schroedinger-s-cat-an-epic-poem).\n", "- What is an example of a classical observable that is not a quantum-mechanical observable? \n", "- Give an example of two quantum-mechanical operators that commute. What about two operators that don't commute? \n", "\n", "## 🤔 Thought-Provoking Questions\n", "- Can you imagine a case where the Hermitian nature of the Hamiltonian operator would be useful? \n", "- Can you imagine a way to use the variational principle in practice?\n", "- The variational principle allows one to choose a wavefunction that gives the lowest energy, which provides the best wavefunction in an energetic sense. This isn't the same as finding the closest wavefunction to the ground-state wavefunction (see below) but if your energy is close enough to the ground-state energy, you can be assured that the wavefunction is also very close to the ground-state wavefunction. Explain why. Assume, for simplicity, that the ground state is nondegenerate.\n", "$$\n", "\\min_\\Psi \\langle \\Psi - \\psi_0 | \\Psi - \\psi_0 \\rangle\n", "$$\n", "\n", "\n", "## 🔁 Recapitulation\n", "- What are the postulates of quantum mechanics?\n", "- Why is it important that quantum-mechanical observables correspond to Hermitian operators?\n", "- What is the Heisenberg Uncertainty principle?\n", "- What is the variational principle?\n", "- List all the Hermitian operators you know.\n", "- What do the Dirac and Kronecker delta notations mean?\n", "\n", "## 🔮 Next Up...\n", "- Multielectron systems\n", "\n", "## 📚 References\n", "My favorite sources for this material are:\n", "- [Randy's book](https://github.com/PaulWAyers/IntroQChem/blob/main/documents/DumontBook.pdf?raw=true)\n", "- D. A. MacQuarrie, Quantum Chemistry (University Science Books, Mill Valley California, 1983)\n", "- [Mathematical Features of Quantum Mechanics](https://github.com/PaulWAyers/IntroQChem/blob/main/documents/PinBox.pdf?raw=true) (my notes, starting page 6).\n", "\n", "There are also some excellent wikipedia articles:\n", "- [Postulates of Quantum Mechanics](https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#Postulates_of_quantum_mechanics)\n", "- [Variational Principle](https://en.wikipedia.org/wiki/Spectral_theory#Spectral_theorem_and_Rayleigh_quotient)" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.3" }, "toc": { "base_numbering": "5", "nav_menu": {}, "number_sections": true, "sideBar": false, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": true, "toc_position": {}, "toc_section_display": false, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 5 }