A string theory

Classical Mechanics Level 3

Suppose a piece of string is drawn taut between two endposts, and let $\phi(x,t)$ denote the height of the string above the line connecting the endposts at the point $x$ along the same line at time $t$ . Which of the following differential equations correctly describes the evolution of the string shape over time?

Assumptions

The composition of the string is uniform.

\frac{\partial^2 \phi}{\partial t^2} \propto \frac{\partial^2 \phi}{\partial x^2}

\frac{\partial^2 \phi}{\partial t^2} \propto \frac{\partial \phi}{\partial x}

\frac{\partial \phi}{\partial t} \propto \frac{\partial^2 \phi}{\partial x^2}

\frac{\partial \phi}{\partial t} \propto \frac{\partial \phi}{\partial x}

1 solution

Michael Mendrin
Jul 6, 2015

Josh, how come the Schrodinger WAVE equation looks like the heat equation, which has this form $\frac{\partial \phi}{\partial t} \propto \frac{\partial^2 \phi}{\partial x^2}$

That's a great question that will take some time for a decent write up. I'll put it here tonight, or make a note, or something else. Stay tuned.

Josh Silverman Staff - 5 years, 11 months ago

Yeah, it's going to take time "for a decent write up". It's an excellent way to show how quantum mechanics differs from classical mechanics, in spite of the fact many undergraduate students miss this fine point. I wish textbooks on QM would spend more time stressing this difference.

Take your time.

Michael Mendrin - 5 years, 11 months ago

A string theory

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