Introduction to Stationary States in Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. One of the key concepts in this discipline is the idea of stationary states. These are specific states of a system whose observable properties do not change with time. In this article, we delve into the relationship between stationary states and the Hamiltonian operator, which plays a central role in understanding how quantum systems evolve over time.

The Definition of Stationary States

In the context of quantum mechanics, a stationary state is a state of a quantum system that is independent of time. Mathematically, this concept can be expressed as a quantum state whose observables do not change with time. A stationary state is often represented by a wave function that is time-independent, and it can be written as:

#931;_t e^{-i E t / h-bar} #931;_0

Here, #931;_0 is the spatial part of the wave function, E is the energy eigenvalue, and h-bar is the reduced Planck's constant. The time evolution of a stationary state can be described by a phase factor, and this phase factor is a function of the energy associated with the state.

The Hamiltonian Operator and Stationary States

The Hamiltonian operator, denoted by H, is a mathematical operator that represents the total energy of a system. It plays a crucial role in the time-dependent Schr?dinger equation:

H #931;t E #931;t

This equation suggests that a stationary state #931;t is an eigenstate of the Hamiltonian operator with the corresponding eigenvalue E. Therefore, all stationary states are eigenstates of the Hamiltonian.

Time-Dependent and Stationary States

For a quantum system with a time-independent potential, the time-dependent wave equation can be separated into the time-dependent and time-independent parts. This separation allows for the formulation of a general solution of the form:

#931;r,t #931;r e^{-i Et / h-bar}

The constant E in this expression arises from the eigenvalue equation for the Hamiltonian operator:

H #931;r E #931;r

Here, #931;r is the eigenfunction. As a result, all expectation values and the probability density calculated using this form of the wave function are time-independent, characterizing the state as a stationary state.

Time Evolution and Stationary States

Another way to understand stationary states is through the time evolution of the wave function. In quantum mechanics, the time evolution of a state is described by the time evolution operator:

#931;t e^{i H t} #931;_0

A stationary state is one that does not change its projection operator over time. This condition is satisfied when:

#931;t #937;t #931;_0

where #937;t is a complex scalar of norm 1, which can be expressed as:

#937;t e^{i #945;t}

Differentiating both sides of this equation with respect to time, one can show that the eigenvalue of the Hamiltonian associated with a stationary state is given by:

H #931;_0 #945;'_0 #931;_0

This confirms that the stationary state #931;_0 is indeed an eigenstate of the Hamiltonian with eigenvalue #945;'_0.

Conclusion

In conclusion, all stationary states in quantum mechanics are eigenstates of the Hamiltonian operator. This relationship is crucial for understanding the dynamics of quantum systems and their time evolution. The Hamiltonian operator encapsulates the total energy of the system, and the eigenstates of this operator correspond to the allowed energy levels of the system.

Further Reading

For a deeper understanding of these concepts and their implications, you may explore:

The Hamiltonian (quantum mechanics) on Wikipedia. Quantum Mechanics textbooks such as Principles of Quantum Mechanics by R. Shankar or Modern Quantum Mechanics by J.J. Sakurai and Jim Napolitano.