Exploring the Differences Between Stationary States and Eigenstates of a Hamiltonian in Quantum Mechanics

Quantum mechanics often deals with states of systems that can be described in terms of their energy levels and the corresponding wave functions. Understanding the differences between stationary states and eigenstates of a Hamiltonian in a quantum system is crucial for a deeper comprehension of quantum theory. This article aims to clarify these concepts and their significance in quantum mechanics.

Introduction to Stationary States and Eigenstates

In quantum mechanics, the state of a system is described by a wave function, which encodes all the information about the system. The wave function is an evolving function, meaning it can change over time, reflecting the dynamic nature of the system. However, in certain situations, the system may exhibit behavior that does not involve any change over time. These are known as stationary states. Stationary states are those states in which the system is at rest, and thus, the expectation values of all the observables (e.g., position, momentum) do not change over time.

Stationary States in Quantum Mechanics

Stationary states are particularly important in the context of time-independent quantum systems. In such systems, the Hamiltonian (the operator corresponding to the total energy of the system) is time-independent. The time-independent Schr?dinger equation for such systems can be written as:

Here, (H) is the Hamiltonian, (psi) is the wave function of the system, and (E) is the energy eigenvalue. The solution to this equation gives us the eigenstates of the Hamiltonian, which correspond to the stationary states of the system. The eigenstates are the special states that, when the system is in one of these states, the expectation value of the energy does not change over time. Thus, eigenstates of the Hamiltonian are also stationary states.

The Relationship Between Stationary States and Eigenstates of the Hamiltonian

The key difference between the terms "stationary state" and "eigenstate of the Hamiltonian" arises from the context and the way they are applied. In a time-independent Hamiltonian system, the state of the system at any time can be expanded as a linear combination of the eigenstates of the Hamiltonian. This is summarized by the time-dependent Schr?dinger equation:

where (hbar) is the reduced Planck's constant. For eigenstates (psi_n) of the Hamiltonian, the evolution of the system is given by:

This shows that eigenstates of the Hamiltonian remain stationary states, as the probability distribution of the system's energy does not change over time.

Examples and Applications

One of the most classic examples to illustrate the difference between stationary states and eigenstates is the particle in a box problem. In this simple model, a particle is confined to a one-dimensional box of length (L), and the Hamiltonian for the particle is:

The eigenstates of the Hamiltonian for this system are:

with corresponding energy eigenvalues:

These eigenstates are stationary states because the particle remains in a specific energy level, and its behavior does not change over time. However, the system is not entirely stationary; the particle can transition between these energy levels, a process governed by quantum mechanics' principles of quantum tunneling and probability transitions.

Conclusion

In conclusion, the concepts of stationary states and eigenstates of the Hamiltonian are fundamental to understanding the behavior of quantum systems. In time-independent Hamiltonian systems, stationary states and eigenstates of the Hamiltonian are essentially the same, as they both describe the states in which the system's energy does not change over time. However, the detailed nature and implications of these states can provide deep insights into the dynamic behavior of quantum systems.

Understanding these concepts is crucial not only for theoretical physicists and quantum mechanics researchers but also for applications in fields such as quantum computing, quantum cryptography, and quantum chemistry. The insights gained from studying these states can lead to cutting-edge technological advancements and scientific breakthroughs.