Question

What is the de Broglie wavelength of a red blood cell, with mass 1.00 $\times$ 10${}^{-11}$ g, that is moving with a speed of 0.400 cm/s? Do we need to be concerned with the wave nature of the blood cells when we describe the flow of blood in the body?

Short answer

The de Broglie wavelength is $1.66\times {10}^{-17}$ m; it's too small to affect blood flow.

Step-by-step solution

Convert Units

First, we need to convert the mass of the red blood cell from grams to kilograms and the speed from centimeters per second to meters per second, as these are the standard units used in physics. Mass: $$1.00\times {10}^{-11}\text{ g}=1.00\times {10}^{-14}\text{ kg}$$Speed: $$0.400\text{ cm/s}=0.004\text{ m/s}$$

Use de Broglie Wavelength Formula

The de Broglie wavelength $\lambda$ is given by the formula:$$\lambda =\frac{h}{mv}$$where:- $h$ is the Planck constant $6.626\times {10}^{-34}$ J·s- $m$ is the mass in kilograms- $v$ is the velocity in meters per second.

Calculate de Broglie Wavelength

Substitute the values into the de Broglie wavelength formula:$$\lambda =\frac{6.626\times {10}^{-34}\text{ J·s}}{1.00\times {10}^{-14}\text{ kg}\times 0.004\text{ m/s}}$$Calculate:$$\lambda =\frac{6.626\times {10}^{-34}}{4.00\times {10}^{-17}}=1.66\times {10}^{-17}\text{ m}$$

Interpret the Result

The calculated de Broglie wavelength is $1.66\times {10}^{-17}$ m, which is extremely small compared to the size of a red blood cell, which is typically around $6\times {10}^{-6}$ m. This means the wave nature of the red blood cell is negligible in the context of describing blood flow in the body.

Key concepts

Wave Nature

The concept of wave nature in particles, including red blood cells, stems from the de Broglie hypothesis. It proposes that all matter exhibits wave-like properties, not just light. This means that everything, from tiny electrons to larger objects like blood cells, has an associated wavelength, known as the de Broglie wavelength. However, this wave nature becomes significant primarily in microscopic particles.

For everyday, macroscopic objects, the wavelength is so tiny that it's practically nonexistent in comparison to the object's size. Therefore, when considering red blood cells or similar biological particles, their wave nature doesn't notably impact their behavior or flow. Since red blood cells in the bloodstream are quite large compared to their minuscule de Broglie wavelength, the wave nature can typically be ignored when examining blood flow.

Understanding this helps explain why quantum mechanics rarely needs to be applied to biological systems at the level of red blood cells. Their physical size and the de Broglie wavelength difference make their quantum effects negligible in the biological processes they are part of.

Planck Constant

The Planck constant is a fundamental constant in physics that plays a crucial role in the study of quantum mechanics. Its value is approximately $6.626\times {10}^{-34}$ Joule-seconds. This constant is incredibly tiny, making it a bridge between the particle and wave nature of matter. In the de Broglie wavelength formula, the Planck constant facilitates transforming momentum into wavelength, giving insight into how properties typically associated with waves, like wavelength, can apply to particles.

When calculating the de Broglie wavelength of a red blood cell, the Planck constant helps determine whether its wave nature matters in practical scenarios, such as in the flow of blood. Because the constant is so small, objects with larger mass and slower speeds result in nearly infinitesimal wavelengths, which explains why even though matter has wave properties, they are generally not observed in larger objects.

The Planck constant is one of the keystones of quantum physics, highlighting how classical mechanics gives way to quantum mechanics at microscopic scales. It serves as a reminder that while the quantum effects are fascinating, they often remain imperceptible in the everyday world of larger bodies, like human cells.

Blood Flow

Blood flow within the human body is a complicated and critical process, governed by principles of fluid dynamics rather than quantum mechanics. In the context of larger biological structures, such as organs and blood vessels, classical physics provides an accurate and straightforward description. Red blood cells are carried along in the bloodstream due to pressures and forces within the circulatory system which ensures oxygen and nutrients are efficiently distributed throughout the body.

The wave nature of individual red blood cells, while interesting from a theoretical perspective, does not have a noticeable effect on the way blood moves. When calculated, the de Broglie wavelength of such cells turns out to be incredibly small compared to their physical size, confirming that their behavior in the bloodstream is more accurately described by classical physics principles.

Therefore, while quantum mechanics offers profound insights into the nature of matter at microscopic levels, it plays little role in macroscopic processes like blood flow. The movement of blood is mostly determined by factors like heart rate, blood pressure, and the geometry of blood vessels, rendering the quantum wave nature irrelevant in practical medical diagnostics and treatments.