Uploaded June 2025 | Updated September 2026, 1 day ago
📘 Lesson 12 of 21
▶️ Watch the full lecture series here: youtube.com/playlist?list=PLIGRVb_-L3Eokg9bMKOX6XLyVgxlPb8uv
Why can't a quantum harmonic oscillator ever be truly at rest?
In this lesson, we revisit the concept of zero-point energy—the minimum possible energy a quantum harmonic oscillator can have. While earlier we saw this arise in a coherent state, here we show that it is actually more general—a direct consequence of the Heisenberg uncertainty principle. Even near equilibrium, quantum fluctuations in position and momentum prevent the total energy from reaching zero.
📝 Solved exercises for this course are available here: quantphys.com/lessons/qm1
📘 Lesson 12 of 21
▶️ Watch the full lecture series here: youtube.com/playlist?list=PLIGRVb_-L3Eokg9bMKOX6XLyVgxlPb8uv
Why can't a quantum harmonic oscillator ever be truly at rest?
In this lesson, we revisit the concept of zero-point energy—the minimum possible energy a quantum harmonic oscillator can have. While earlier we saw this arise in a coherent state, here we show that it is actually more general—a direct consequence of the Heisenberg uncertainty principle. Even near equilibrium, quantum fluctuations in position and momentum prevent the total energy from reaching zero.
📝 Solved exercises for this course are available here: quantphys.com/lessons/qm1
![Do Wave Functions Vanish at Infinity? [QBE Ep. 5]](https://i.ytimg.com/vi/t8x4Kw11c80/mqdefault.jpg "Do Wave Functions Vanish at Infinity? [QBE Ep. 5]
Episode 5 of Quantum on the Back of an Envelope [QBE].
🔗 Watch the full series here: https://www.youtube.com/playlist?list=PLIGRVb L3EpDzApE5EnAOYyTAyagvi1W
In this episode of Quantum on the Back of an Envelope, In this episode of Quantum on the Back of an Envelope, we ask whether a quantum mechanical wave function really has to vanish at infinity — an assumption often made in introductory textbooks and courses. To explore this, we examine a specific example (taken from Principles of Advanced Mathematical Physics by Robert D. Richtmyer) of a wave function that is perfectly continuous, differentiable, and normalizable, yet not only refuses to vanish, but actually becomes unbounded at infinity.")
![Attractive Potential with No Ground State [QBE Ep. 2]](https://i.ytimg.com/vi/z6HmvZ70S2A/mqdefault.jpg "Attractive Potential with No Ground State [QBE Ep. 2]
Episode 2 of Quantum on the Back of an Envelope [QBE].
🔗 Watch the full series here: https://www.youtube.com/playlist?list=PLIGRVb L3EpDzApE5EnAOYyTAyagvi1W
In this episode of Quantum on the Back of an Envelope, we explore the strange and fascinating -1/x^2 potential. Despite being attractive and infinitely deep, it has no ground state and therefore no discrete energy spectrum. In a certain regime, the system can be so unstable that it could (theoretically) destroy the universe. No equations just simple dimensional analysis.")
