One Image Summary of Quantum Mechanics

A 'Spherical Cow' in a Simple Harmonic Oscillator Potential is pretty much Quantum Mechanics in a nutshell.

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Image generated using ChatGPT-5 on 2025-Oct-19.

Random musing about Classical Projections

Assume I have a measuring device to measure the electric field intensity $\vec{E}$ in a specific direction, say, $z$. If $\vec{E}$ is oriented at an angle $\theta$ to the measuring device (or, equivalently to the $z$ direction), then the measuring device will register an intensity of $E\cos\theta$,  $E$ being the magnitude of $\vec{E}$. So far this is in line with classical physics and the parallelogram law of vector addition.

However, no measuring device is instantaneous. Every measuring device takes a finite amount of time to interact with the quantity-under-measurement before settling to a final value. This may not seem terribly relevant as the combined system comprising the measuring device and quantity-under-measurement converges to a steady state in short order. In classical physics, this steady state is real and the measuring device is truly measuring the underlying quantity.

A different perspective

One could posit a different take on the above measurement process: Since the measurement takes a finite time, one could argue that even in the steady state a measuring device is measuring a range of values - in rapid fashion - and is only able to present an average value as (macroscopic) measuring devices simply cannot respond fast enough.

Here's how one might do this. Note that

$$E\cos\theta = E\left[\cos^2(\frac{\theta}{2}) - \sin^2(\frac{\theta}{2})\right] \\ = E\cos^2(\frac{\theta}{2}) + (-E) \sin^2(\frac{\theta}{2})$$

So far this is just mathematical manipulation. However, this new form lets us look at the measurement process from a different perspective.

Perhaps the measuring device only ever measures $+E$ or $-E$ but with respective probabilities $P(+E) = \cos^2(\frac{\theta}{2})$ and $P(-E) = \sin^2(\frac{\theta}{2})$. Note that the individual probabilities add up to $1$, and the average value of $E$

$$ \langle E \rangle = E\cos^2(\frac{\theta}{2}) + (-E) \sin^2(\frac{\theta}{2}) = E\cos\theta$$

is exactly what's presented as the measured value by the measuring device.

Your point being ... ?

It may seem silly to look at it this way but it forms a nice bridge to the discussion of spin-$\frac{1}{2}$ in quantum physics. The only difference being that an individual spin-$\frac{1}{2}$ measurement yields a specific value (either $+\hbar/2$ or $-\hbar/2$) as it's the result of an instantaneous interaction between a Stern-Gerlach apparatus and the spin-$\frac{1}{2}$ particle under measurement. However, over an ensemble, the individual measurements yield the same average (‘expectation value’) as the classical case above.

Uniqueness of Ladder Operators in the Harmonic Oscillator

Background

In introductory Quantum Mechanics textbooks, the algebraic method of determining the energy eigenstates of the harmonic oscillator involves factorising the Hamiltonian as $$H=\hbar\omega(a^\dagger a + \frac{1}{2})$$ and showing that the $a^\dagger$ and $a$ operators respectively raise and lower the energy by $\hbar\omega$. Thus, the spectrum of the harmonic oscillator is determined to be a ladder with a step-size of $\hbar\omega$.

However, there is rarely any discussion on why this is the end all and be all for the harmonic oscillator spectrum. How do we know that there aren't other operators or, equivalently, a different factorisation of $H$, that would yield a new ladder with a possibly different step-size. This question had been nagging me for a long time. Now, with the help of ChatGPT-5, I understand why.

Uniqueness of ladder operators

The harmonic oscillator Hamiltonian is $$H = \frac{p^2}{2m} + \frac{1}{2}m \omega^2 x^2$$ with corresponding operator identities $$[x,p] = i\hbar, \quad [H,x] = -\frac{i \hbar}{m} p, \quad [H,p] =  i m \omega^2 \hbar x.$$

Assume a general operator $$B = \alpha x + \beta p$$ in the linear span of $\{x,p\}$ and demand that it be an eigenoperator of the adjoint action $ad_H(\cdot) = [H,\cdot]$ of $H$. In other words $$[H,B] = \lambda B$$ which implies $B$ acts a ladder with fixed energy step $\lambda$.

Evaluating the commutator relationship on the left hand side gives $$[H,B] = i\hbar(\beta m \omega^2 x - \frac{\alpha}{m}p).$$

Equating coefficients on the right hand side of the above two equations we get $$i \hbar \beta m \omega^2 = \lambda \alpha, \quad -\frac{i\hbar}{m}\alpha = \lambda\beta.$$

Nontrivial solutions, with $\alpha$ and $\beta$ not both $0$, exist only if $\lambda^2 = (\hbar\omega)^2$, so $\lambda = \pm\hbar\omega$. Thus, at least within the linear span of $\{x,p\}$, there are exactly two nontrivial eigenoperators of the adjoint action $ad_H(.)$ with eigenvalues $\pm\hbar\omega$. The corresponding normalised operators, unique up to a phase, turn out to be the canonical ladder operators $$a^\dagger = \sqrt{\frac{m\omega}{2\hbar}}x - i\frac{p}{\sqrt{2m\hbar\omega}}$$ and $$a = \sqrt{\frac{m\omega}{2\hbar}}x + i\frac{p}{\sqrt{2m\hbar\omega}}$$ and they shift energies by $\pm\hbar\omega$. 

(Justification for the above claim: for $\lambda = \pm\hbar\omega$, we get $\beta = \mp\frac{i}{m\omega}\alpha$ and hence $B = \alpha(x \mp \frac{i}{m\omega}p)$. Equivalently, $B=\alpha\sqrt{\frac{2\hbar}{m\omega}}a^{\dagger}$ or $B=\alpha\sqrt{\frac{2\hbar}{m\omega}}a.$  To wit, $B$ is proportional to $a^\dagger$ and $a$.)

Multiplying a ladder $B$ with any function $f(H)$, which commutes with $H$, produces another operator with the same step size $\lambda$: $$ [H,f(H)B] = \lambda f(H)B. $$

Handy Summary of Classical Mechanics

Key references

1. Goldstein, Herbert; "Classical Mechanics"; 1959; Addison-Wesley Publishing Company, Inc.; archive.org link to the 3rd edition (2002).
2. Lanczos, Cornelius; "The Variational Principles of Mechanics"; 4th ed.; 1970; Dover Publications, Inc..
3. Taylor, John R.; "Classical Mechanics"; 2005; University Science Books.
4. Susskind, Leonard & Hrabovsky, George; "The Theoretical Minimum: What You Need to Know to Start Doing Physics"; 2013; Basic Books.
5. Coopersmith, Jennifer; "The Lazy Universe"; 2017; Oxford University Press.
6. Goldstine, Herman H.; "A History of the Calculus of Variations from the 17th through the 19th Century"; 1980; Springer-Verlag;  archive.org link.

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Lagrangian mechanics

Hamilton's principle

The actual path which a holonomic system follows between two points $1$ and $2$  in configuration space in a given time interval, $t_1$ to $t_2$, is such that the action integral

$$\begin{equation}\label{eqn:action:functional}S = \int_{t_1}^{t_2} \mathcal{L}dt\end{equation}$$

is stationary when taken along the actual path.

$\mathcal{L(\mathbf{q},\dot{\mathbf{q}},t)}$ is the Lagrangian, a function of the generalised coordinates $\mathbf{q}$, generalised velocities $\dot{\mathbf{q}}$, and time $t$, and satisfies the Euler-Lagrange equations

$$\begin{equation} \frac{d}{dt}\frac{\partial{\mathcal{L}}}{\partial{\dot{q_i}}} - \frac{\partial \mathcal{L}}{\partial q_i} = 0 \quad [i=1, \dots, n] \end{equation}$$

$n$ being the number of degrees of freedom.

The generalized force $F_i$ and generalized (or conjugate) momentum $p_i$ are respectively given by $$\begin{equation} \frac{\partial{\mathcal{L}}}{\partial q_i} = F_i, \quad \frac{\partial{\mathcal{L}}}{\partial \dot{q_i}} = p_i\end{equation}$$

The Euler-Lagrange equations can be rewritten as $F_i = \dot{p_i}$.

Historical Note

The Euler-Lagrange equations were first derived by Euler using a geometric method. They were subsequently refined by Lagrange using what is now known as The Calculus of Variations and d'Alembert's Principle of Virtual Work. The modern presentation based on the action integral is essentially the reformulation by Hamilton when he introduced his Principle of Least Action.

The coinage "Calculus of Variations" is due to Euler after he was impressed by Lagrange's analytical work and preferred it over his own method.

Refer Goldstine's text for more on the historical development.

Conservation and symmetry

If the Lagrangian does not contain a coordinate $q_i$ (though it may contain $\dot{q_i}$) then the coordinate is said to be ignorable and the corresponding conjugate momentum $p_i$ is conserved.

Conservation of Hamiltonian

If $\mathcal{L}$ does not depend explicitly on time ($\partial{\mathcal{L}}/\partial{t} = 0$), then the Hamiltonian

$$\begin{equation}\mathcal{H}(\mathbf{q},\mathbf{p},t) = (\sum_{i=1}^{n}\dot{q_i} p_i) - \mathcal{L}\end{equation}$$

is conserved.

For most systems, $\mathcal{H}$ is just the total energy and is frequently conserved. However, the identification of $\mathcal{H}$ as a constant of motion and as the total energy are two separate matters, and the conditions sufficient for one are not enough for the other.

Lagrangian for electromagnetism

The Lagrangian for a charge $q$ with mass $m$ in an electromagnetic field is $$\begin{equation}\mathcal{L}(\mathbf{r}, \dot{\mathbf{r}}, t) = \frac{1}{2}m\dot{\mathbf{r}}^2 - q(V-\dot{\mathbf{r}}\cdot\mathbf{A})\end{equation}$$

$V(\mathbf{r},t)$ and $\mathbf{A}(\mathbf{r},t)$ being the scalar and vector potentials respectively.

Physics and Intuition

One common criticism of modern Physics (i.e., Relativity and Quantum Mechanics) is that it is counter-intuitive. This is not unexpected: our evolutionary history in no way requires that we have an innate understanding of the mechanisms underpinning the Universe at all scales. That said, this criticism is not uniquely limited to modern Physics. In this post I explore concepts that we often take for granted today, and how those could have seemed counter-intuitive to people of earlier times.

Time Dilation Again - Much Ado about Nothing?

Preliminaries

The Lorentz transforms in the Theory of Special Relativity (SR) between frames O and O' defined in the usual way are given by

t' = L (t - vx)

x' = L (x - vt)

where v is the relative velocity of the origin of O' with respect to the origin of O (along the x-direction), L = (1 - v^2)^(-1/2) > 1, and t is measured in terms of distance so that the velocity of light, c, becomes equal to 1.

Time Dilation

The usual treatment of Time Dilation is as follows. For a clock situated at the origin of O' we have x'=0, so that

x = vt

and

t' = L (1-v^2) t = t/L

Since L > 1, this means that the clock at the origin of O' runs slower by a factor of L compared to the clock in the frame of O which is coincident with it. This has been called Time Dilation. And the usual interpretation is that the clocks in the frame of O' actually tick slower than those in O. This is further taken to mean that O' physically ages slower than O.

What does the origin of O see?

Consider the perspective of the observer located at the origin of O. She sees clocks of O' going past her with speed v in the x-direction. At any given instant t, at the origin of O (i.e. x=0) we get,

t' = Lt

This means that the clocks of O' coincident with the origin of O run faster by exactly the same factor L. (Ultimately, this is to be expected due to the symmetry of the Lorentz transforms and the Principle of Relativity.)

Now, the observer located at the origin of O would conclude that the clocks of O' run faster than her clock. This is at variance with the observer located at x(=vt) in O who, as we have seen earlier, concludes that the clocks of O' run slower in comparison. In fact, different observers located at different points in the frame of O all come to different conclusions about the rate at which clocks in O' tick. Interestingly, there is a locus of events given by

x = t (L - 1) / (Lv)

where the clocks of O and O' agree. This locus lies between the world lines of O and O'.

Nowhere have I seen the argument that the observations of the observer at the origin of O should be used to conclude that the exact opposite of Time Dilation takes place!

What does all this really mean?

Ultimately, this boils down to the fact that frames O and O' don't agree on the assignment of space-time coordinates to space-time events. According to O, the clocks of O' are not synchronized correctly. And according to O', the clocks of O are not synchronized correctly. However, neither O nor O' can argue that the intrinsic rate at which their individual clocks tick vary merely due to the relative velocity between the two frames. This also follows from the Principle of Relativity because neither frame is privileged. In fact, time as a measurement of physical/dynamical/physiological duration has to tick at the same rate in both frames. This is because the Lorentz transforms are kinematical and do not include any dynamical effects that would affect the physiological rate of ageing. At best, the Lorentz transforms should be interpreted as scaling transforms that maintain covariance between two inertially moving frames (i.e. the speed of light, c, remains 1 in both frames).

Demolishing "Paradoxes" in Special Relativity: Twins in a Cylindrical Minkowskian Universe

Introduction

The kinematics of Special Relativity (SR) is predicated on the Lorentz transforms [Lorentz1952]. Although these equations are merely (rescaling) transforms that conserve the covariance of physical laws across relatively moving inertial frames, it is the general perception that the effect of time-dilation (derived from the Lorentz transforms) is a manifest physical effect [1]. As any student of SR knows, this interpretation leads to various paradoxes; notably, the Twin (or Clock) Paradox which is the focus of discussion here [2].

The Twin Paradox

Very simply, the Twin Paradox involves two identical twins, one of whom stays on Earth (frame O) while the other, an astronaut (frame O'), goes on a space-faring journey in a high-speed rocket. Upon her return, the astronaut twin ostensibly finds that she has aged less than her sister who stayed back on Earth - presumably due to effects of time-dilation. This is clearly a paradox because the astronaut twin could very well argue that it's her sister on Earth who should age less, due to the very same effects of time-dilation.

The paradox has been explained away in multiple ways, but mostly by invoking the fact that the situation is not completely symmetrical. For instance, only the astronaut twin actually experiences acceleration when she turns around. Alternatively, the astronaut twin uses two inertial frames - one in each direction of travel - and it's the switch between the frames that causes the asymmetry [Schutz1985].

I submit that all the explanations of the Twin Paradox are faulty; primarily because they seek to explain something that is not a paradox in the first place. And to demonstrate this, I have to re-frame the problem in a way that eliminates the underlying basis for asymmetry.

Anisotropic Wavefront in Special Relativity - Comment on Vankov's "On Controversies in Special Relativity"

Prologue
I recently read Anatoli Andrei Vankov's paper "On Controversies in Special Relativity" [Van2006] with great interest. I've always been fascinated by discussions of paradoxes in Relativity (especially, the Ehrenfest Paradox), but I wasn't in the least aware that the Spherical Wave-front example of Einstein was controversial.

Vankov's Analysis
See Section 3 (Shape of light front, and constancy of the speed of light) of Vankov's paper [Van2006] for

1. Specifics on why the example is controversial.
2. An analysis of the physics underlying the example.
   
3. Subsequent interpretation towards the resolution of the controversy.

Specifically, the analysis concludes that the shape of an isotropic (spherical) wave-front in frame S' becomes an ellipsoid in frame S. I do not disagree with the analysis itself. However, I submit that the eventual interpretation is untenable. [There are a few typos in equations (11), (13), and (14). However, the typos don't affect the analysis perse; so I don't dwell on them here.]

Note that we start off with a spherical wave-front in frame S' which is a surface of constant t'. This surface is then parametrized using theta'. Subsequently, using the Lorentz transforms, this spherical surface is determined to be transformed into an ellipsoid in frame S. And the analysis is absolutely correct.

My Comments
However, at this point it is also asserted that an ellipsoid is exactly what is perceived by frame S. But, it must be realized that this ellipsoid surface is not a surface of constant t in frame S (because of relativity of simultaneity). [This is also evident in the analysis: The expression for the time coordinate in frame S is a function of theta/theta'.] As a result, the conclusion that the ellipsoid surface is perceived as-is in frame S, is, philosophically speaking, untenable: If frame S were to specify the shape of the wave-front, it would be based on a surface of constant t, and not based on a surface that corresponds to space-time events belonging to varying, or even arbitrary, t. Thus, the ellipsoid surface has no significance in frame S.

From a physical perspective, frame S would assert that the wave-front is spherical because frame S will make the shape determination at a specific time t which would give an obviously spherical wave-front (just as Einstein had originally remarked). Note that the space-time events that make up this spherical wave-front in frame S similarly carry no significance when transformed to frame S'.

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Update [15Sep2009]: I've emailed a one-page write-up to Anatoli Vankov soliciting his comments.

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References

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[Van2006] Vankov, "On Controversies in Special Relativity", arXiv.org, 2006.

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Relativity of Simultaneity - Bad Explanations on the Web

A common example used to demonstrate the Relativity of Simultaneity goes like this, more or less:

A train of length D is passing by a platform of length D. An observer P stands in the middle of the platform, and there's another observer T who sits in the middle of the train. P notices that the front end of the train coincides with the front edge of the platform AT THE SAME TIME as when the rear end of the train coincides with the rear edge of the platform. The two 'coincidence events' are thus simultaneous in P's frame of reference.

The question is, does T agree that both 'coincidence events' are simultaneous in T's frame of reference? (It is implicitly assumed that this question is posed in P's frame of reference and that both reference frames are inertial.)

The correct answer in Special Relativity is "No".

Call the front coincidence event F and the rear coincidence event R. Then, according to T, F happens before R. The events are NOT simultaneous in T's frame of reference even though they are simultaneous in P's frame of reference.

The Bad Explanation

Often times, the bad explanation for this is an incomplete version of a correct explanation provided by Einstein himself [Ein1924]. The bad explanation goes as follows: