##### Department of Mathematics,

University of California San Diego

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### Final Defense

## Pun Wai Tong

#### UCSD

## Classical Limit on Quantum Mechanics for Unbounded Observables

##### Abstract:

Let $p_{\hbar}$ and $q_{\hbar}$ be momentum and position operators respectively. In 1973, Hepp showed the classical limit and quantum correction of an observable $e^{i(rq_{\hbar}+sp_{\hbar})}$ under the evolution generated by a Hamiltionian \[ H_{\hbar}=-\frac{\hbar}{2m}\partial^2+V\left(\sqrt{\hbar}x\right) \] on all state $\psi\in L^{2}\left(\mathbb{R}\right)$ in The Classical Limit for Quantum Mechanical Correclation Functions. In contrast to the Hepp' s result, in our talk, we are interested in unbounded ``observables'' and more general Hamiltionians. Motivated by the idea in Quantum Fluctuations and Rate of Convergence towards Mean Field Dynamics by Rodnianski and Schlein in 2009. Classical mechanics can be recovered from quantum mechanics by studying the asymptotic behavior of quantum expectations relative to $\sqrt{\hbar}.$

Advisor: Bruce Driver

### March 4, 2016

### 12:00 PM

### AP&M 6402

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