Quantum free fall
master's thesis
The title of my thesis was Weak equivalence principle in the quantum regime: Compatibility with quantum mechanics, and tunneling effects via the different quantizations of the time of arrival operator. The weak equivalence principle (WEP) in the quantum regime has been the subject of many studies with a broad range of approaches to the problem. In the thesis, we tackle the problem anew using a time-operator based theory of quantum time of arrival.
Classically, the time of arrival at the origin for a non-relativistic point particle that is projected upward with initial velocity \(v = v_o\) at an initial postion \(q = -q_o\) is given by
\[t_{\pm} = \dfrac{v_o}{g} \left( 1 \pm \sqrt{1-2g \dfrac{q_o}{v_o^2}} \right),\]where \(g\) is the acceleration due to gravity. The toa \(t_{-}\) indicates the first time crossing of the particle at the origin while \(t_{+}\) indicates the second time crossing as the particle falls down after reaching the classical turning point.
Various TOA operators were constructed using supraquantization, and quantization by Weyl, Born-Jordan, and simple symmetric ordering rules. The expectation value of the TOA operators were explicitly shown to be equal to the classical time of arrival plus mass-dependent quantum correction terms, implying incompatibility of the WEP. The full extent of the violation of the WEP was shown via the mass dependence of the TOA distribution.
I was able to publish a paper on Physical Review A and two conference proceedings from this thesis. I also presented part of this thesis during the poster session of the conference Time and Fundamentals of Quantum Mechanics which was held at the Weizmann Institute of Science.
