What is the Difference Between h and h-bar

The main difference between h and h-bar is that h is the Planck constant, whereas ℏ (pronounced “h-bar”) is the reduced Planck constant, defined as h/2π.

The Planck constant (h) and the reduced Planck constant (ħ) are fundamental constants in quantum mechanics and are often used in equations related to the behavior of particles at the quantum level. They play a crucial role in describing the quantization of energy levels and the wave-particle duality of matter.

Key Areas Covered

1. What is h  
      – Definition, Features, Role
2. What is h-bar
      – Definition, Features 
3. Similarities Between h and h-bar
      – Outline of Common Features
4. Difference Between h and h-bar
      – Comparison of Key Differences
5. FAQ: h and h-bar
      – Frequently Asked Questions

Key Terms

h, h-bar, Reduced Planck Constant, Planck Constant

What is h

Planck’s constant, denoted by the symbol “h,” is a fundamental constant in quantum mechanics. It plays a pivotal role in understanding the behavior of particles at the atomic and subatomic levels. Planck’s constant is a proportionality factor that relates the energy of a photon to its frequency. The equation below expresses this relationship

E = hν

where E is the photon’s energy, h is Planck’s constant, and ν is the frequency of the radiation.

This simple equation laid the groundwork for quantum theory, challenging classical physics notions that could not explain certain phenomena at the atomic scale.

The value of Planck’s constant is approximately 6.626 x 10^^-34 joule seconds. It is an exceedingly small number, underscoring the quantum nature of reality. The fact that energy is quantized in discrete packets, or quanta, rather than being continuous, as classical physics suggested, is a revolutionary concept that Planck’s constant helped to establish.

Role of h

One of the critical areas where Planck’s constant comes into play is in understanding blackbody radiation. Classical physics struggled to explain the radiation spectrum emitted by a heated object, such as a metal. However, Planck proposed a new theoretical approach that involved quantizing the energy levels of the oscillators responsible for emitting radiation. By introducing the concept of energy quantization, he derived a formula that perfectly described the observed blackbody radiation spectrum. This achievement marked the birth of quantum theory.

Figure 1: Black Body Radiation

Furthermore, Planck’s constant is integral to Werner Heisenberg’s uncertainty principle. The uncertainty principle states that it is impossible to simultaneously know both the precise position and momentum of a particle. The product of the uncertainties in position (Δx) and momentum (Δp) is greater than or equal to Planck’s constant divided by 2π: ΔxΔp ≥ h/2π. This principle fundamentally challenges the determinism inherent in classical physics, highlighting the inherent probabilistic nature of quantum systems.

Planck’s constant is also central to understanding wave-particle duality. The duality concept suggests that particles, such as electrons, exhibit both wave-like and particle-like behaviors. Building on Planck’s ideas, Louis de Broglie proposed that the wavelength (λ) associated with a particle is inversely proportional to its momentum, with Planck’s constant as the proportionality factor: λ = h/p. This relationship laid the groundwork for the development of quantum mechanics and complemented the evolving understanding of the atomic world.

The value of h defines the scale of quantum effects in the physical world. For instance, when considering the sizes of atoms and the magnitudes of quantum phenomena, Planck’s constant sets the boundary between classical and quantum realms. Classical physics provides an accurate description at scales much larger than the Planck constant, while at smaller scales, quantum mechanics takes precedence.

What is h-bar

h-bar is reduced Planck constant (ħ). The reduced Planck constant (ħ) is derived from Planck’s constant (h). It is defined as ħ = h / (2π). This adjustment by 2π arises due to the frequent appearance of the factor (2π) in quantum equations. The introduction of ħ simplifies many formulas, making them more elegant and compact.

In quantum mechanics, a particle’s angular momentum is quantized and given by the expression L = nħ, where n is an integer. This quantization of angular momentum is a fundamental aspect of the quantum nature of particles.

Similarities Between h and h-bar

• Both ℎ and ℏ originate from Max Planck’s work on quantum mechanics.

Difference Between h and h-bar

Definition

h is the original Planck constant, while h-bar is its reduced form, scaled by 2π.

Nature

The Planck constant is a fundamental constant that relates the energy of a quantum system to the frequency of its associated electromagnetic wave. Reduced Planck constant is commonly used in quantum mechanics to simplify mathematical expressions, especially in relation to angular momentum and certain wave functions.

FAQ: h and h-bar

Is the h-bar a constant?

Yes, ħ (h-bar) is a constant. It is the reduced Planck constant and represents a fundamental value in quantum mechanics.

What does the h-bar equal in quantum mechanics?

The currently accepted value for ℏ is 1.054571817×10^-34 J·s (according to the 2019 International Bureau of Weights and Measures).

How do you convert H to h-bar?

To convert the Planck constant “H” to the reduced Planck constant “ħ” (h-bar), you divide “H” by 2π. It’s like taking the regular Planck constant and adjusting it by a factor of 2π to get the reduced version.

Conclusion

h is the Planck constant, a fundamental constant in quantum mechanics, while h-bar is the reduced Planck constant, obtained by dividing “h” by $2\pi$. Thus, this is the main difference between h and h-bar.

Reference:

1. “Planck’s Constant.” Encyclopedia Britannica. 

Image Courtesy:

1. “Black body” By Darth Kule – Own work (Public Domain) via Commons Wikimedia