Heisenberg Picture Example

Delving into the Heisenberg Picture: A Practical Approach to Quantum Mechanics

Quantum mechanics presents two primary formulations: the Schrödinger picture and the Heisenberg picture. While the Schrödinger picture depicts the evolution of the system's state vector over time, the Heisenberg picture focuses on the time-dependent evolution of the operators. This article aims to clarify the Heisenberg picture, explaining its core concepts, contrasting it with the Schrödinger picture, and illustrating it with practical examples. We will see how this seemingly different perspective offers equivalent yet insightful ways to understand quantum dynamics.

Understanding the Core Concepts

In the Schrödinger picture, the state vector |Ψ(t)> changes with time according to the time-dependent Schrödinger equation, while operators remain time-independent. The Heisenberg picture, conversely, holds the state vector constant in time (equal to its initial value |Ψ(0)>) while the operators evolve. This seemingly abstract shift offers a powerful perspective, especially when dealing with conserved quantities.

The crucial equation governing the time evolution of an operator Â in the Heisenberg picture is:

dÂ/dt = (i/ħ)[H, Â] + ∂Â/∂t

where:

H is the Hamiltonian of the system (representing its total energy).
ħ is the reduced Planck constant.
[H, Â] is the commutator of the Hamiltonian and the operator Â, defined as [H, Â] = HÂ - ÂH.
∂Â/∂t represents any explicit time dependence in the operator itself (independent of the Hamiltonian).

This equation reveals that the time evolution of an operator depends on its commutator with the Hamiltonian. If the operator commutes with the Hamiltonian ([H, Â] = 0), then the operator is a constant of motion (like energy, momentum, or angular momentum in certain systems).

Schrödinger vs. Heisenberg: A Comparative Glance

The key difference lies in what evolves with time. In the Schrödinger picture, the state vector evolves, describing how the system's state changes. Measurements are performed on this evolving state. In the Heisenberg picture, the state vector remains fixed at its initial value, while the operators representing physical quantities evolve. The expectation value of an observable remains the same regardless of which picture we use. This ensures both formulations offer equivalent descriptions of the quantum system. The choice of picture often depends on the specific problem and what aspect of the system is being emphasized.

Example: The Simple Harmonic Oscillator

Let's consider a one-dimensional quantum harmonic oscillator. The Hamiltonian is given by:

H = (p²/2m) + (1/2)mω²x²

where:

p is the momentum operator.
m is the mass.
ω is the angular frequency.
x is the position operator.

Using the Heisenberg equation of motion, we can find the equations of motion for x and p. The commutator [x, p] = iħ is crucial here. After calculating the commutators and solving the resulting differential equations, we obtain the time evolution of the position and momentum operators:

x(t) = x(0)cos(ωt) + (p(0)/mω)sin(ωt)
p(t) = p(0)cos(ωt) - mωx(0)sin(ωt)

These equations describe the oscillatory behaviour of the position and momentum operators in the Heisenberg picture. Notice that the initial values x(0) and p(0) remain constant.

A More Complex Example: Spin in a Magnetic Field

Consider a spin-1/2 particle in a constant magnetic field along the z-axis. The Hamiltonian is given by:

H = -μ · B = -γB Sz

where:

μ is the magnetic moment.
B is the magnetic field.
γ is the gyromagnetic ratio.
Sz is the z-component of the spin operator.

Applying the Heisenberg equation of motion, the time evolution of the spin operators can be derived. We will find that Sz is a constant of motion, as expected, since it commutes with the Hamiltonian. However, Sx and Sy will evolve in time, exhibiting precession around the z-axis.

Conclusion

The Heisenberg picture provides an alternative, yet equivalent, framework for analyzing quantum systems. By focusing on the time evolution of operators, it highlights the dynamical aspects of physical quantities and offers a particularly elegant perspective for understanding conserved quantities. While the Schrödinger picture might be intuitively easier to grasp initially, the Heisenberg picture is indispensable for more advanced quantum mechanical calculations and provides a deeper insight into the fundamental nature of quantum dynamics. Choosing the appropriate picture depends heavily on the problem at hand and the desired perspective.

FAQs

1. What is the main advantage of the Heisenberg picture over the Schrödinger picture? The Heisenberg picture is advantageous when dealing with conserved quantities, as they appear as time-independent operators. It's also useful in situations where the Hamiltonian is time-dependent, as it can simplify calculations.

2. Is the expectation value of an observable the same in both pictures? Yes, the expectation value of an observable remains identical in both the Schrödinger and Heisenberg pictures.

3. Can the Heisenberg picture be used for systems with time-dependent Hamiltonians? Yes, the Heisenberg picture can handle time-dependent Hamiltonians, although the calculations can become more complex.

4. How does the Heisenberg picture relate to quantum field theory? The Heisenberg picture forms a foundation for quantum field theory, where field operators evolve in time.

5. Are there any limitations to the Heisenberg picture? While powerful, the Heisenberg picture can be less intuitive for beginners compared to the Schrödinger picture, and dealing with complicated Hamiltonians can lead to challenging calculations.

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Heisenberg Picture Example

Delving into the Heisenberg Picture: A Practical Approach to Quantum Mechanics

Understanding the Core Concepts

Schrödinger vs. Heisenberg: A Comparative Glance

Example: The Simple Harmonic Oscillator

A More Complex Example: Spin in a Magnetic Field

Conclusion

FAQs

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