Day: March 17, 2022

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202203171422 Half Picture of Quantum Mechanics 01B

(This half picture is based on the second half of manuscript of Chapter One, 2015-2016 PHYS3351 Quantum Mechanics Lecture Notes.)

Section 01

Momentum

As the wavefunction describes the state of a quantum system, how do we find the information of velocity or momentum?

Expectation value \langle\mathbf{r}\rangle of position \mathbf{r}

\langle \mathbf{r}\rangle=\displaystyle{\iiint\mathrm{d}V\,\mathbf{r}|\Psi (\mathbf{r},t)|^2}

is a function of time t.

\begin{aligned} \frac{\mathrm{d}\langle\mathbf{r}\rangle}{\mathrm{d}t} & = \iiint\mathrm{d}V\,\mathbf{r}\frac{\partial}{\partial t}|\Psi |^2 \\ & = \frac{i\hbar}{2m}\iiint\mathrm{d}V\,\mathbf{r}\nabla\cdot [\Psi^*\nabla\Psi - (\nabla\Psi^* )\Psi ] \\ \dots &\textrm{ integration by parts }\dots \\ \frac{\mathrm{d}\langle\mathbf{r}\rangle}{\mathrm{d}t} & = -\frac{i\hbar}{2m}\iiint [\Psi^*\nabla\Psi - (\nabla\Psi^* )\Psi ]\,\mathrm{d}V \\ \dots &\textrm{ integration by parts }\dots \\ \underbrace{\frac{\mathrm{d}\langle\mathbf{r}\rangle}{\mathrm{d}t}}_{\textrm{velocity}} & =\iiint\mathrm{d}V\,\Psi^*\bigg( -\frac{i\hbar}{m}\nabla\bigg)\Psi \\ \end{aligned}

\displaystyle{\frac{\mathrm{d}\langle \mathbf{r}\rangle}{\mathrm{d}t}} is the “velocity” of the central position of the particle “cloud”; whereas

\displaystyle{m\frac{\mathrm{d}\langle\mathbf{r}\rangle}{\mathrm{d}t}=\iiint\mathrm{d}V\,\Psi^*(-i\hbar\nabla )\Psi}

is a very suggestive form for us to introduce the momentum operator:

\displaystyle{\hat{p}=\frac{\hbar}{i}\nabla}.

Expectation value:

\begin{aligned} \langle\hat{p}\rangle & = -i\hbar\iiint\mathrm{d}V\,\Psi^*\nabla\Psi \\ & = m\frac{\mathrm{d}\langle \mathbf{r}\rangle}{\mathrm{d}t} \\ \end{aligned}

Kinetic energy:

\hat{T}=\displaystyle{\frac{\hat{p}}{2m}=-\frac{\hbar^2}{2m}\nabla^2}

Angular momentum:

\hat{L}=\mathbf{r}\times \hat{p} = \displaystyle{\mathbf{r}\times \bigg(\frac{\hbar}{i}\nabla\bigg)}.

\begin{aligned} \langle\hat{T}\rangle & = -\frac{\hbar^2}{2m}\iiint\mathrm{d}V\,\Psi^*\nabla^2\Psi \\ \langle \hat{L}\rangle & = \iiint\mathrm{d}V\,\Psi^*\bigg(\mathbf{r}\times\frac{\hbar}{i}\nabla \bigg)\Psi \\ \end{aligned}

Summary.

In quantum mechanics, all observable quantities, such as momentum, position, or energy, are described by operators. When we describe a particle by the wavefunction \Psi (\mathbf{r},t), we are looking at the probability distribution at the position representation. In this representation the position operator takes the special form of a number: \mathbf{r}; whereas other observables such as momentum and energy involve the differential operator \nabla. For a particle described by a wavefunction, the expectation value of the observed quantity is given by

\langle\hat{S}\rangle = \displaystyle{\iiint\mathrm{d}V\,\Psi^*\hat{S}\Psi}

where \hat{S} denotes the operator form of the observable.

Section 02

Uncertainty principle

Like the standard deviation of position distribution, we can also evaluate the standard deviation of momentum:

\sigma_{p_x}^2=\langle\hat{p}_x^2\rangle - \langle\hat{p}_x\rangle^2.

For any possible wavefunction, one finds that

\sigma_x\sigma_{p_x}\geqslant\displaystyle{\frac{\hbar}{2}}.

Example: Gaussian wavepacket

\begin{aligned} \Psi (x,0) & = A\exp \bigg(-\frac{x^2}{4\sigma^2}+iKx\bigg) \\ A & = (2\pi\sigma^2)^{-1/4} \\ \hat{p}_x & = \frac{\hbar}{i}\frac{\partial}{\partial x} \\ \Psi (x,0) & = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\phi (k)e^{ikx}\,\mathrm{d}k \\ \phi (k) & = (2\sigma^2/\pi )^{1/4}\exp [-\sigma^2(k-K)^2] \\ \end{aligned}

\begin{aligned} \langle \hat{p}_x^n\rangle & = \int_{-\infty}^{\infty}\bigg[ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\phi^*(k)e^{-ikx}\,\mathrm{d}k\bigg](\hat{p}_x)^n\bigg[ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\phi (k')e^{ik'x}\,\mathrm{d}k'\bigg] \\ & = \int_{-\infty}^{\infty}\mathrm{d}x\,\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\phi^* (k)e^{-ikx}\,\mathrm{d}k\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\phi (k')(\hbar k')^ne^{ik'x}\,\mathrm{d}k' \\ & = \int_{-\infty}^{\infty}\mathrm{d}k\,\phi^* (k)\int_{-\infty}^{\infty}\phi (k')\delta (k-k')(\hbar k')^n\,\mathrm{d}k' \\ & = \int_{-\infty}^{\infty}(\hbar k)^n|\phi (k)|^2\,\mathrm{d}k \\ \end{aligned}

so

\begin{aligned} \langle\hat{p}_x\rangle & = \int\mathrm{d}x\,\Psi^*\bigg( \frac{\hbar}{i}\frac{\partial}{\partial x}\bigg)\Psi = \hbar K \\ \Delta p_x & = \langle \hat{p}_x^2\rangle - \langle \hat{p}_x\rangle^2 = \frac{\hbar}{2\sigma} \\ \dots \textrm{ because }&\Delta x=\sigma \\ \Delta x\cdot\Delta p_x & = \frac{\hbar}{2} \\ \end{aligned}

\therefore Gaussian wavepackets satisfy the uncertainty relation. This relation tells us that the position and the momentum of a particle can \textrm{\scriptsize{NEVER}} be simultaneously determined with precision. This also shows the significance of the constant \hbar. Only when we talk about quantities with this scale, quantum effect is significant, that is, things cannot be described by classical mechanics. But if we cannot measure things with such precision, you won’t “notice” the quantum effect: that is why our daily experience is classical.

Von Neumann projection postulate:

Measurement collapses the wavefunction.

E.g., A position measurement with the outcome \mathbf{r}_0 collapses the wavefunction from \Psi (\mathbf{r},t) to \delta (\mathbf{r}-\mathbf{r}_0). So a second measurement immediately following will also find the particle at \mathbf{r}_0.

Note the subtle difference of expectation value from mean value in the usual sense.

(to be continued)

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202203170840 Half Picture of Quantum Mechanics 01A

(This half picture is based on the first half of manuscript of Chapter One, 2015-2016 PHYS3351 Quantum Mechanics Lecture Notes.)

Section 01

Given a particle of mass m and a force F(x,t), we classify

\begin{cases}\textrm{conservative }F =\displaystyle{-\frac{\partial V(x)}{\partial x}}\enspace\big( V(x)\textrm{: potential energy}\big)\\ \textrm{nonconservative} \\ \end{cases}

In classical mechanics, description of dynamics is given by

i. trajectory (i.e., position as a function of time):

x(t)

ii. velocity:

\mathbf{v}=\displaystyle{\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}}

iii. momentum:

p=m\displaystyle{\frac{\mathrm{d}x}{\mathrm{d}t}}

iv. kinetic energy:

T=\displaystyle{\frac{1}{2}m\bigg(\frac{\mathrm{d}x}{\mathrm{d}t}\bigg)^2}

v. solving from Newton’s second law \textrm{Net }F=ma:

\displaystyle{F=ma\qquad\Longrightarrow\qquad m\frac{\mathrm{d}^2x}{\mathrm{d}t^2}=F=-\frac{\partial V}{\partial x}}

In quantum mechanics, description of dynamics is given by wavefunction \Psi (x,t) which is a single-valued, finite, and continuous complex function.

Statistical interpretation.

|\Psi (x,t)|^2\,\mathrm{d}x: the probability of particle being between x and x+\mathrm{d}x at time t

In 3D case, |\Psi (x,y,z,t)|^2\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z: the probability of particle being in the volume between x and x+\mathrm{d}x, y and y+\mathrm{d}y, and z and z+\mathrm{d}z.

\Psi (x) is also known as the probability amplitude.

Imagine the particle as a “cloud” which is moving and changing its shape, instead of a point object following a trajectory curve in space. The particle has a larger probability to be at the denser point of the “cloud”. Even if you know everything about the particle dynamics, that is, the wavefunction \Psi (\mathbf{r},t), you cannot predict with certainty the outcome of an experiment to measure its position. Instead, you can say the probability you will detect the particle in the volume between x and x+\mathrm{d}x, y and y+\mathrm{d}y, and z and z+\mathrm{d}z is

|\Psi (x,y,z,t)|^2\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z.

Normalization.

\displaystyle{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\, |\Psi (x,y,z,t)|^2=1}

Q. Which one of the following is \textrm{\scriptsize{NOT}} a possible wavefunction in the region x\geqslant 0 for \beta >0:

(I) Axe^{\beta x}; (II) Axe^{-\beta x}; (III) Ae^{-\beta x}.

A. (I)

Important characteristics of a wavefunction

A discrete example

There are a number of positions x_1, x_2, … ,x_{10} on a 1D line, at which the particle can appear with probability \rho_1, \rho_2, … , \rho_{10}.

Normalization: \rho_1 + \rho_2+\cdots +\rho_{10}=1

If we throw dice for N\gg 1 times, each time we get a particle at one of the positions:

x(j)\in\{ x_1, x_2, \dots ,x_{10}\} for j=1,2,\dots ,N

the times we get the particle at position x_i is given by N\rho_i,

i.e., the number of x(j)\big|_{j=1,\dots ,N}=x_i is N\rho_i.

Average position:

\begin{aligned} \langle x\rangle & = \sum_{j=1}^{N}x(j)\frac{1}{N}\\ & = \sum_{i=1}^{10}\frac{x_iN\rho_i}{N} \\ & = \sum_{i=1}^{10}x_i\rho_i \\ \end{aligned}

Numerical measure of the amount of spread:

\begin{aligned} \Delta x(j) & \equiv x(j)-\langle x\rangle \\ \sum_{j}\Delta x(j) & = 0 \\ \langle (\Delta x)^2\rangle & \equiv \sum_{j}\frac{1}{N}(\Delta x(j))^2 \\ & = \sum_{j}\frac{1}{N}(x^2(j)+\langle x\rangle^2-2x(j)\langle x\rangle )\\ & = \sum_{j}\frac{1}{N}x^2(j) - \langle x\rangle^2 \\ & = \langle x^2\rangle -\langle x\rangle^2 \textrm{ (always +ve)} \end{aligned}

Owing to the statistical description of quantum mechanics, we can no longer speak of the precise position of a particle, nevertheless, we still care about two quantities: (I) the expectation value of position; and (II) the uncertainty in position.

Expectation value is the weighted average of all possible values that a physical observable can take.

Expectation value of position:

\langle \mathbf{r}\rangle \equiv \displaystyle{\iiint\mathrm{d}V\,\mathbf{r}|\Psi (\mathbf{r},t)|^2}

Deviation.

\begin{aligned} \Delta\mathbf{r} & = \mathbf{r}-\langle \mathbf{r}\rangle \\ \langle \Delta\mathbf{r}\rangle & \equiv \langle\mathbf{r}\rangle - \langle\langle\mathbf{r}\rangle\rangle =0 \\ \langle (\Delta\mathbf{r})^2\rangle & = \langle\mathbf{r}^2-2\mathbf{r}\langle \mathbf{r}\rangle + \langle \mathbf{r}\rangle^2\rangle \\ & = \langle \mathbf{r}^2\rangle - \langle\mathbf{r}\rangle^2 \\ \end{aligned}

For the x-, y-, z-components of position, observe that

\begin{aligned} \langle (\Delta x)^2\rangle & = \langle x^2\rangle - \langle x\rangle^2 \\ \langle (\Delta y)^2\rangle & = \langle y^2\rangle - \langle y\rangle^2 \\ \langle (\Delta z)^2\rangle & = \langle z^2\rangle - \langle z\rangle^2 \\ \langle (\Delta\mathbf{r})^2\rangle & = \langle (\Delta x)^2\rangle + \langle (\Delta y)^2\rangle + \langle (\Delta z)^2\rangle \\ \end{aligned}

Standard deviation in position (aka uncertainty)

\sigma is defined by \sigma^2=\langle (\Delta\mathbf{r})^2\rangle;

The expectation value of position \langle \mathbf{r}\rangle gives the central position of the particle “cloud”, and \sigma tells how much the “cloud” spreads out in space. Roughly speaking, you will find the particle in the space interval of

(\langle x\rangle\pm\sigma_x, \langle y\rangle\pm\sigma_y, \langle z\rangle\pm\sigma_z)

Example (Gaussian wavepacket)

Consider a wavefunction

\Psi (x,0)=A\exp \bigg( \displaystyle{-\frac{x^2}{4\sigma^2}+iKx} \bigg).

By the normalization requirement,

\begin{aligned} \int\mathrm{d}x\, |\Psi (x,0)|^2 & = 1 \\ |A|^2\int\mathrm{d}x\,\exp\bigg( -\frac{x^2}{2\sigma^2}\bigg) & = 1 \\ A & = (2\pi\sigma^2)^{1/4} \end{aligned}

\begin{aligned} \langle x\rangle & = \int_{-\infty}^{\infty}\mathrm{d}x\, x(2\pi\sigma^2)^{-1/2}\exp \bigg( -\frac{x^2}{2\sigma^2}\bigg) = 0 \\ \langle x^2\rangle & = \int_{-\infty}^{\infty}\mathrm{d}x\, x^2(2\pi\sigma^2)^{-1/2}\exp \bigg( -\frac{x^2}{2\sigma^2}\bigg) \\ & = \int_{-\infty}^{\infty}\frac{1}{2}\,\mathrm{d}x^2\, x(2\pi\sigma^2)^{-1/2}\exp \bigg( -\frac{x^2}{2\sigma^2}\bigg) \\ & = \int_{-\infty}^{\infty}(-\sigma^2)\,\mathrm{d}\bigg(\exp\bigg( -\frac{x^2}{2\sigma^2} \bigg)\bigg)x(2\pi\sigma^2)^{-1/2} \\ & = \int_{-\infty}^{\infty}\sigma^2(2\pi\sigma^2)^{-1/2}\exp\bigg( -\frac{x^2}{2\sigma^2} \bigg)\,\mathrm{d}x \\ & = \sigma^2 \\ \Rightarrow \langle (\Delta x)^2\rangle & = \sigma^2 \\ \end{aligned}

Section 02

The dynamics of the wavefunction \Psi (\mathbf{r},t) is described by Schrödinger equation:

\displaystyle{i\hbar\frac{\partial \Psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\Psi + V(\mathbf{r},t)\Psi = \hat{H}\Psi}.

Hamiltonian operator:

\hat{H}=\hat{T}+\hat{V}

Kinetic energy operator:

\displaystyle{\hat{T}=-\frac{\hbar^2}{2m}\nabla^2}

Potential energy operator:

\hat{V}=V(\mathbf{r},t)

Planck constant:

\hbar =1.0546\times 10^{-34}\,\mathrm{J\, s}

Given the wavefunction \Psi (\mathbf{r},t_0) at time t_0, the Schrödinger equation determines \Psi (\mathbf{r},t) for all future time.

What is quantum mechanics? The simplest answer is wavefunction (and its statistical interpretation) plus Schrödinger equation.

In integral form, Schrödinger equation reads:

\Psi (x,t)=\Psi (x,t_0)\underbrace{\exp \bigg(-i\frac{\hat{H}}{\hbar}(t-t_0) \bigg)}_{\textrm{evolution operator}}

Probability density and flux

According to the probability interpretation, wavefunction has to be normalized. In a 1D example:

\displaystyle{\int_{-\infty}^{\infty}|\Psi (x,t)|^2\,\mathrm{d}x=1}

Motivation.

Does \Psi (x,t) remain normalized in the evolution according to Schrödinger equation? I.e.,

\displaystyle{\frac{\mathrm{d}}{\mathrm{d}t}\int_{-\infty}^{\infty}|\Psi (x,t)|^2\,\mathrm{d}x\stackrel{?}{=}0}

The answer to this question is affirmative.

Proof.

To begin with,

\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\int_{-\infty}^{\infty}|\Psi (x,t)|^2\,\mathrm{d}x & = \int_{-\infty}^{\infty}\bigg[ \frac{\partial}{\partial t}|\Psi (x,t)|^2\bigg]\,\mathrm{d}x \end{aligned}

evaluating the square-bracketed term in the integrand,

\displaystyle{\frac{\partial}{\partial t}|\Psi (x,t)|^2=\Psi^*\frac{\partial \Psi}{\partial t}+\frac{\partial\Psi^*}{\partial t}\Psi}

by identities of Schrödinger’s

\begin{aligned} \frac{\partial \Psi}{\partial t} & = \frac{i\hbar}{2m}\frac{\partial^2\Psi}{\partial x^2} - \frac{i}{\hbar}V\Psi\\ \frac{\partial \Psi^*}{\partial t} & = -\frac{i\hbar}{2m}\frac{\partial^2\Psi^*}{\partial x^2} +\frac{i}{\hbar}V\Psi^*\\ \end{aligned}

it follows that

\begin{aligned} \frac{\partial}{\partial t}|\Psi |^2 & = \frac{i\hbar}{2m}\bigg( \Psi^*\frac{\partial^2\Psi}{\partial x^2}-\frac{\partial^2\Psi^*}{\partial x^2}\Psi \bigg) \\ & = \frac{\partial}{\partial x}\bigg[ \frac{i\hbar}{2m}\bigg( \Psi^*\frac{\partial\Psi}{\partial x}-\frac{\partial\Psi^*}{\partial x}\Psi \bigg) \bigg]\\ \end{aligned}

and

\displaystyle{\frac{\mathrm{d}}{\mathrm{d}t}\int_{-\infty}^{\infty}|\Psi (x,t)|^2\,\mathrm{d}x = \frac{i\hbar}{2m}\bigg( \Psi^*\frac{\partial\Psi}{\partial x} - \frac{\partial\Psi^*}{\partial x}\Psi\bigg)\bigg|_{-\infty}^{\infty}}

since \Psi (x,t) is normalized, i.e.,

\Psi (x=\pm\infty ,t)=0,

so

\displaystyle{\frac{\mathrm{d}}{\mathrm{d}t}\int_{-\infty}^{\infty}|\Psi (x,t)|^2\,\mathrm{d}x=0}.

\therefore The Schrödinger equation guarantees the wavefunctions remain normalized.

QED

Defining the probability density by:

\rho (x,t)\equiv |\Psi (x,t)|^2

and the probability current by:

\displaystyle{j=-\frac{i\hbar}{2m}\bigg(\Psi^*\frac{\partial \Psi}{\partial x} - \frac{\partial\Psi^*}{\partial x}\Psi\bigg)}

we write the continuity equation:

\displaystyle{\frac{\partial}{\partial t}\rho =-\frac{\partial}{\partial x}j}.

Note that

\underbrace{\frac{\partial}{\partial t}\int_{a}^{b}\rho\,\mathrm{d}x}_{\textrm{(A)}}= \underbrace{-j(b)}_{\textrm{(B)}} +\underbrace{j(a)}_{\textrm{(C)}}

Physical meanings of (A), (B), and (C):

\textrm{(A)}: change in the probability of being between a and b;
\textrm{(B)}/\textrm{(C)}: positive (/negative) value means inward (/outward) flux into the region between a and b.

In 3D case:

\begin{aligned} \mathbf{j}(\mathbf{r}) & =-\frac{i\hbar}{2m}[\Psi^*\nabla\Psi - (\nabla\Psi^*)\Psi ] \\ \frac{\partial}{\partial t}|\Psi |^2 & = -\nabla\cdot\mathbf{j} \\ \end{aligned}

Probability flux through a surface area is \mathbf{j}\cdot\mathrm{d}\mathbf{s}.

(to be continued)