WignerFunction - crowlogic/arb4j GitHub Wiki
What is the Wigner Function?
The Wigner function is a type of distribution function used in quantum mechanics. It's a representation of a quantum state in phase space, which is a conceptual framework that combines position and momentum space.
Purpose and Applications:
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Bridge Between Classical and Quantum Physics: The Wigner function is particularly notable because it helps bridge the gap between classical and quantum physics. It represents quantum states in a way that resembles classical probability distributions in phase space, but with key quantum characteristics.
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Quantum Mechanics Analysis: It is used to analyze quantum systems in a way that is somewhat analogous to how classical statistical mechanics uses phase space distributions.
Key Features:
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Quasi-probability Distribution: The Wigner function is not a true probability distribution because it can take negative values, something that is not possible for classical probability distributions. These negative values are indicative of inherently quantum phenomena, like quantum interference.
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Calculation of Observables: It can be used to calculate the expectation values of quantum observables.
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Visualization of Quantum States: The Wigner function provides a way to visualize quantum states in phase space, offering insights into their properties that might not be as apparent in other representations.
Mathematical Formulation:
- The Wigner function $W(x, p)$ for a quantum state described by a wave function $\psi(x)$ in one dimension is given by:
$$W(x, p) = \frac{1}{\pi \hbar} \int_{-\infty}^{+\infty} \psi^*(x + y) \psi(x - y) e^\frac{2ipy}{\hbar} dy$$
where $x$ is the position, $p$ is the momentum, and $\hbar$ is the reduced Planck constant.
Interpretation Challenges:
- Due to its ability to assume negative values, interpreting the Wigner function as a probability distribution in the classical sense is problematic. These negative regions are interpreted as manifestations of quantum "weirdness," like entanglement and superposition.
The Wigner function $W(x, p)$ has several distinct properties that make it a unique and valuable tool in quantum mechanics, particularly in the phase space analysis of quantum systems. These properties stem from its role as a quasi-probability distribution and its relationship to the underlying quantum state. Here are some key properties:
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Real-valued: Despite being associated with quantum mechanics, the Wigner function is always a real function, not a complex one. This means for any given point in phase space, $W(x, p)$ has a real value.
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Normalization: When integrated over the entire phase space, the Wigner function yields the total probability, which is always 1 for a normalized quantum state. Mathematically, this is expressed as:
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} W(x, p) dx dp = 1$$
- Marginal Distributions: The marginal distributions of the Wigner function reproduce the probability distributions of position and momentum in quantum mechanics. Specifically:
- Integrating $W(x, p)$ over all momenta gives the position probability density:
$$\int_{-\infty}^{\infty} W(x, p) dp = |\psi(x)|^2$$
- Integrating over all positions gives the momentum probability density:
$$\int_{-\infty}^{\infty} W(x, p) dx = |\phi(p)|^2$$
where $\psi(x)$ is the wave function in position space and $\phi(p)$ is the wave function in momentum space.
- Symmetry: The Wigner function is symmetric under a simultaneous sign change of both position and momentum:
$$W(-x, -p) = W(x, p)$$
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Limits to Classical Probability: In the classical limit (where quantum effects become negligible), the Wigner function converges to a classical probability distribution in phase space.
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Non-Positivity: Unlike classical probability distributions, the Wigner function can take on negative values. These negative values are indicative of quantum interference and are a hallmark of quantum mechanical phenomena.
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Covariance under Phase Space Transformations: The Wigner function transforms in a covariant manner under canonical transformations of phase space. This means that its form is preserved under shifts in position and momentum, and under rotations in phase space.