From Classical Mechanics to Quantum Theory

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 2025/06/25   Share

Why the derivative of kinetic energy with respect to velocity is momentum

One of the most elegant and profound identities in classical physics is the fact that the derivative of kinetic energy with respect to velocity gives the momentum:

$\frac{d}{dv} \left( \frac{1}{2}mv^2 \right) = mv = p$

On the surface, this might seem like a mere coincidence—a fortunate mathematical convenience. But as we explore this identity from the perspectives of Newtonian mechanics, Lagrangian mechanics, Hamiltonian formalism, and finally quantum mechanics, we’ll uncover a deep unifying principle in physics. This blog post aims to walk you through this connection with clarity and conceptual rigor.

1. Newtonian Mechanics: A Direct Calculation

We start with the classical expression for kinetic energy:

$E_k = \frac{1}{2}mv^2$

Differentiating with respect to velocity:

$\frac{dE_k}{dv} = mv = p$

This straightforward computation gives the linear momentum of the object. In this view, momentum is simply a measure of how sensitive the kinetic energy is to changes in velocity.

But this is not just a numerical trick—it hints at a deeper structure in mechanics, where kinetic energy and momentum are dual aspects of motion.

2. Lagrangian Mechanics: Momentum as a Fundamental Definition

In Lagrangian mechanics, the dynamics of a system are described by the Lagrangian:

$L(q, \dot{q}, t) = T - V = \frac{1}{2}m\dot{q}^2 - V(q)$

Here, $q$ is the generalized coordinate and $\dot{q}$ the generalized velocity.

The generalized momentum is defined as:

$p = \frac{\partial L}{\partial \dot{q}}$

For our simple system:

$\frac{\partial L}{\partial \dot{q}} = m\dot{q} = p$

✅ This means the identity $\frac{dE_k}{dv} = p$ is built into the Lagrangian formalism itself—momentum is defined as the rate of change of the Lagrangian (which contains the kinetic energy) with respect to velocity.

Moreover, this definition is not restricted to linear systems or Cartesian coordinates. It applies to any coordinate system and any type of generalized velocity.

3. Hamiltonian Mechanics: Duality of Momentum and Velocity

We move on to Hamiltonian mechanics, where we switch from $(q, \dot{q})$ to $(q, p)$ as the fundamental variables.

The Hamiltonian is defined via a Legendre transform:

$H(q, p, t) = p\dot{q} - L(q, \dot{q}, t)$

For many mechanical systems, $H$ turns out to be the total energy:

$H = T + V = \frac{p^2}{2m} + V(q)$

The Hamilton’s equations govern time evolution:

$\dot{q} = \frac{\partial H}{\partial p}, \quad \dot{p} = -\frac{\partial H}{\partial q}$

From the first equation:

$\dot{q} = \frac{p}{m} \Rightarrow p = m\dot{q}$

✅ This shows that velocity is the derivative of energy with respect to momentum, the dual of the Newtonian and Lagrangian statement.

In Lagrangian mechanics: $p = \frac{\partial T}{\partial \dot{q}}$
In Hamiltonian mechanics: $\dot{q} = \frac{\partial H}{\partial p}$

This duality is a hallmark of symplectic geometry, the mathematical foundation of Hamiltonian mechanics.

4. Quantum Mechanics: Momentum as a Generator of Translations

In quantum mechanics, momentum is no longer just $mv$ . It becomes an operator:

$\hat{p} = -i\hbar \frac{\partial}{\partial x}$

This operator generates translations in space, just as energy (the Hamiltonian) generates translations in time.

Recall the Schrödinger equation:

$\hat{H}\psi = i\hbar \frac{\partial \psi}{\partial t}$

This duality of energy/time and momentum/space echoes the classical ideas:

Classical: $p = \frac{dT}{dv}$
Quantum: $\hat{p} = -i\hbar \partial_x$ , and acts on wavefunctions

Moreover, the quantum kinetic energy operator is:

$\hat{T} = \frac{\hat{p}^2}{2m} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2}$

So if you apply the operator $\hat{T}$ to a plane wave $\psi(x) = e^{ikx}$ , its eigenvalue is $\frac{\hbar^2k^2}{2m}$ , and the momentum is $\hbar k$ . Again, this reproduces the classical identity in operator form.

5. Final Reflection: A Deep Unity

This journey—from Newton to Lagrange, Hamilton, and Schrödinger—reveals that the identity:

$\frac{dT}{dv} = p$

is far more than a curious derivative. It is a manifestation of the deep connection between energy, motion, and symmetry.

In Newtonian mechanics, it’s a calculational result.
In Lagrangian mechanics, it’s a definition.
In Hamiltonian mechanics, it’s a dual variable.
In quantum mechanics, it’s an operator.

The unity of these views showcases the power of physics: different languages describing the same fundamental truths.

🌟 Bonus Question for Reflection

In special relativity, does $\frac{dE}{dv} = p$ still hold?

(Spoiler: No, but it leads to even deeper insights about mass-energy equivalence and relativistic dynamics.)

Author: Xinchuan Du
Write at National University of Singapore during FLEPS 2025 conference

Author：Du Xinchuan

原文链接：http://www.icalculate.website/2025/06/25/From-Classical-Mechanics-to-Quantum-Theory/

发表日期：June 25th 2025, 4:44:11 pm

更新日期：June 26th 2025, 10:01:03 am

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Deck of Doctoral Dissertation

CATALOG

1. 1. Newtonian Mechanics: A Direct Calculation
2. 2. Lagrangian Mechanics: Momentum as a Fundamental Definition
3. 3. Hamiltonian Mechanics: Duality of Momentum and Velocity
4. 4. Quantum Mechanics: Momentum as a Generator of Translations
5. 5. Final Reflection: A Deep Unity
6. 🌟 Bonus Question for Reflection

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