I am applying to MIT for admission to the Ph.D. program in Physics, a critical step towards becoming a successful researcher in this field. Condensed Matter Physics is my main research interest, particularly in method development of quantum many-body theory and its applications where many-body interactions and quantum fluctuation play an important role in molecules, liquids or solids.
Joining the Ph.D. program in Physics at Duke is a critical step for me toward a successful theoretical researcher. With my talents in Theoretical Physics, particularly in method development and numerical modeling to tackle quantum many-body problems, I am eagerly looking forward to doing research with brilliant colleagues and professors in your program.
同時也把theoretical physics 點出來，會更確切。
With my talents 聽起來比較厲害些。
最後的 looking forward to ... 加上去我覺得意念上比較完整。
Even though I knew I wanted to be a physicist before I started at XXX University, it was my master’s degree work at the Institute of Atomic and Molecular Sciences at Academia Sinica that developed my research skills, and, most importantly, my resolution to pursue a career as a physics researcher. The work on quantum many-body interactions in hydrogen-bonded systems has not only focused my research interests in theoretical and computational physics but also led my exploration into modern advances in condensed matter physics. As I deeply aspire to become a successful research scientist in condensed matter physics, MIT is my top choice for doctoral study.
After the solid undergraduate Physics training at XXX University, it is my master’s degree work at the Institute of Atomic and Molecular Sciences (IAMS), Academia Sinica, Taiwan, that fully invigorated my confidence and resolution to pursue a career as a research scientist. The work was primarily focused on a theoretical understanding of quantum many-body interactions in hydrogen bonded systems, and it also led me to explore other related research topics in Theoretical Physics.
During my stay at IAMS, I worked with Prof. XXX and Prof. OOO. My research focus was on the development and applications of numerical methods for gaining an understanding of the quantum many-body interactions inside hydrogen-bonded systems. In the first year, I worked on path integral approaches to study the quantum effects of light nuclei. Based on the imaginary-time path integral formalism, I implemented the ab-initio path integral molecular dynamics method to evaluate quantum partition functions via computational simulations. The implementation incorporates state-of-art path integral techniques with electronic structure programs, and I effectively enhance its performance via parallel computing.The inclusion of nuclear quantum effects has been argued to be important in hydrogen-bonded systems, yet it has often been neglected in typical ab-initio methods. This tool now enables our group to carry out full-dimensional quantum simulations on parallel computer clusters to capture nuclear quantum effects at finite temperatures.
In the first year, I worked on path integral approaches to study the nuclear quantum effects in hydrogen-bonded systems. Based on the imaginary-time path integral formalism, I implemented ab-initio path integral molecular dynamics method to quantize the nuclei’s degrees of freedom on a full-dimensional Born-Oppenheimer surface. The inclusion of nuclear quantum effects has been argued important in hydrogen-bonded systems but often neglected in typical ab-initio methods owing to its fundamental and computational difficulties. My implementation incorporated state-of-art path integral techniques with electronic structure programs, and its performance was significantly boosted by my parallel programming which effectively free us from time and size limitations in numerical simulations. This tool now enables our group to carry out full-dimensional quantum simulations on parallel computers to study nuclear quantum effects of hydrogen-bonded systems at finite temperatures.
Later on, I turned my focus towards the theoretical study of vibrational spectroscopy. The research problem that my current work addresses is the difficulty in accurately accounting for vibrational transitions of hydrogen-bonded clusters when the vibration modes are anharmonic and coupled to other degrees of freedom. To this end, we have devised a multi-dimensional nuclear vibrational Hamiltonian under Discrete Variable Representation and obtained corresponding eigenstates via direct diagonalization. By working through a few examples on ionic hydrogen-bonded clusters, we are beginning to shed some light on the coupling mechanism of vibrations and its effects on infrared spectra. Typically, for water in liquid phases, strong absorptions near 3600 cm-1 are assigned to be stretches of the OH group. In our study, we found the frequency of an OH stretch varies across a wide range, from 1000 cm-1 to 3800 cm-1, depending on its surroundings. In the meantime, the excitations of low frequency modes can stimulate strong anharmonic coupling with OH stretches, which can cause peak splitting or broadening in the vibrational spectra. Understanding the concept of vibrational coupling directly helps us to decipher complicated experimental spectra when strong resonance occurs between vibrational transitions.
At the second year, I turned my focus toward the spectroscopic interrogation of hydrogen-bonded clusters in which the biggest challenge was on an accurate account for vibrational transitions when they are anharmonic and coupled to other degrees of freedom. To this end, we have devised a multi-dimensional nuclear vibrational Hamiltonian and through the study on ionic hydrogen-bonded clusters we have shed some light on the coupling mechanism of vibrational transitions in hydrogen-bonded systems. For example, the H2O(aq) strong absorptions near 3600 cm-1 are assigned to be the first transitions of OH stretch group, but in our study we found the transition frequency of OH stretch can vary across a wide range, from 1000 cm-1 to 3800 cm-1, depending on its surroundings. Also, excitations of low frequency modes can stimulate strong anharmonic coupling with OH stretches causing peak splitting or broadening of infrared spectra. Such finding is immensely helpful for deciphering complicated experimental spectra when a strong resonance occurs between vibrational transitions.
Although these are my current areas of specialization, other research topics related to condensed matter also appeal to me. Novel phenomena discovered in liquids and solids are not only challenging our understanding in Quantum Mechanics but also stimulating technological innovations. The Taiwan Semiconductor Manufacturing Company (TSMC) invited me to participate in their Elite Camp and shared with me their technological perspective on solid states. I learned about the major challenges for the next phase of development in lithography, surface reaction and smaller scale semiconductors, all of which rely on future breakthroughs of fundamental physics. Moreover, taking Prof. xxx’s course “Topics on Theoretical Material Physics” has also inspired me to explore ongoing research problems related to quantum many-body theory. It never ceases to amaze me how the interplay between motions of electrons and atoms could render such a rich variety of physical phenomena in condensed matter systems.
Through my exploration into condensed matter physics, many questions have been raised in my mind, including: How and why can collective modes or coherent dynamics of a group of particles occur and bring quantum physics into a macroscopic scale? How can such macroscopic quantum phenomena be understood and harnessed? Moreover, what is the next important physics discovery that could lead us to achieve the control and manipulation of quantum states and quantum phases in condensed matter? I am zealously seeking a clearer and more unified physics picture to begin to answer them.
這段的重點在講原作的Possible future plan. 但是若能跟教授、program的特點作為結合會更好也更精簡。所以這段後來就被刪掉了。
MIT is the best place to continue that path. As my research interests span many subareas of condensed matter physics, I have browsed with curiosity the research interests of Prof. John Joannopoulos (ab-initio methods), Prof. Jeremy England (biophysics), Prof. Mehran Kardar (biophysics & soft matter physics), Prof. Patrick Lee (superconductivity & spin liquid), Prof. Senthil Todadri (phenomena beyond Landau’s theory), and Prof. Xiao-Gang Wen (string-net condensation).
Studying these professors’ research websites and publications gives me the sense that this program is a great match for my interests and will be continuously hallenging and exciting in my future study. With your guidance, I will be able to fully develop and exert my research capability and start contributing to this research field. I expect challenges; attending MIT will provide the best setting to reach my goal of being a successful researcher. I am ready to take this step.
Through my research experience, I especially recognize the underlying importance of Electron Structure Theory; to find a proper description of the interplay between electrons and atoms for understanding properties of materials and chemical reactions. Though great advances in this field have been achieved in these decades, several outstanding problems still have resisted solutions including self-interaction error in Density Functional Theory, determination of HOMO-LUMO gap and large-scale simulations of materials or biomolecules. It is my ambition to work on these and make significant contribution to this field.
Furthermore, in the search of a suitable future graduate program, I am exicted to learn that Prof. Wei-Tao Yang and Prof. Jian-Feng Lu both teach at Duke. The exceptional book, “Density-Functional Theory of Atoms and Molecules”, written by Prof. Yang is the best among all density-functional theory textbooks I have ever read. I anticipate participating in Prof. Yang’s current study on tracing delocalization error and static correlation error, serving as a key role to construct better approximated exchange-correlation. Meanwhile, I am seeking an opportunity to work with Prof. Lu as well. His innovative research on electronic structure theory, multi-scale modeling and sampling of rare events is also at the top of my research interests, and I already have abundant experience in numerical modeling of quantum many body problems.
The Department of Physics at Duke University outstandingly integrates Biophysics, AMO Physics, Condensed Matter Physics, High Energy Physics and other related disciplines together; such as stable algorithm and numerical methods developed in Computational Physics can have great impact on not only lattice QCD calculation in High Energy Physics but also collective quantum phenomena in Condensed Matter Physics. As aspiring to involve cross-disciplinary research, I am deeply attracted by the multi-disciplinary characteristic of your program, and believe your program provides the best environment that continues to nourish my cross-disciplinary research interests.
Finally, I have to thank Prof. XXX’s and Prof. XXX’s mentoring during my master’s study, and the honorable invitation to Elite Camp held by Taiwan Semiconductor Manufacturing Company (TSMC) that further strengthened my resolution to continue my doctoral study in Physics. In the Elite Camp, I realized my gifts in theoretical physics and my potential to contribute to physics academia. With my strong enthusiasm in Physics, I believe I will keep growing and honing my research abilities in your program and become a successful researcher in the future.