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The Variational Principle in Quantum Mechanics

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The Variational Principle in quantum mechanics is a key computational tool for estimating ground state energies when exact solutions are elusive. It relies on the premise that the ground state energy is the lowest a system can have, using trial wave functions to approximate this energy. This principle is not only foundational in quantum physics but also critical in quantum computing, particularly in algorithms like the Variational Quantum Eigensolver (VQE), which are instrumental in solving complex problems in chemistry and materials science.

The Variational Principle in Quantum Mechanics

The Variational Principle is a fundamental concept in quantum mechanics that provides a powerful computational tool for estimating the ground state energy of a quantum system. It is based on the premise that the ground state energy is the lowest possible energy that a system can have. Mathematically, the principle is expressed by the inequality \(E_{0} \leq \frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}\), where \(E_{0}\) is the true ground state energy, \(\psi\) is a trial wave function from the Hilbert space, and \(H\) is the Hamiltonian operator of the system. This inequality implies that the expectation value of the Hamiltonian, calculated with any trial wave function that is normalized, will always be an upper bound to the ground state energy. By optimizing the trial wave function, one can find the best approximation to the ground state energy.
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Applications of the Variational Principle in Quantum Physics

The Variational Principle is a versatile tool in quantum physics, essential for estimating the ground state energies and corresponding wave functions of quantum systems when exact solutions are not possible. It is applicable to both time-independent and time-dependent problems. In the realm of quantum computing, the principle is the foundation of the Variational Quantum Eigensolver (VQE) algorithm, which is designed to find the lowest eigenvalue of a Hamiltonian. This algorithm is particularly promising for quantum information processing, as it can be implemented on near-term quantum computers to solve problems in chemistry and materials science.

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00

Variational Principle application scope

Used for estimating ground state energies/wave functions when exact solutions are unattainable; applies to time-independent/time-dependent problems.

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Variational Quantum Eigensolver (VQE) purpose

VQE algorithm seeks lowest eigenvalue of a Hamiltonian, foundational for quantum computing.

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VQE algorithm significance for near-term quantum computers

Enables solving chemistry/materials science problems on current quantum computers, leveraging the Variational Principle.

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