Quantum Mechanics: De Broglie Wavelength and Uncertainty

De Broglie Wavelength

• De Broglie wavelength is the wavelength associated with a particle's momentum, given by the equation λ = h/p, where h is Planck's constant and p is the momentum.
• The de Broglie wavelength of an electron with a speed of 1 * 10^8 m/s is 7.27 * 10^-12 m.
• The de Broglie wavelength of a 40 keV electron is 6.15 * 10^-12 m.
• The de Broglie wavelength of a 1 mg grain of sand moving at 20 m/s is extremely small, negligible for practical purposes.

Kinetic Energy and de Broglie Wavelength

• The kinetic energy of an electron with a de Broglie wavelength equal to that of a 100 keV x-ray is 100 keV.

Heisenberg's Uncertainty Principle

• Heisenberg's Uncertainty Principle states that the product of the uncertainties in position (Δx) and momentum (Δp) of a particle is always greater than or equal to Planck's constant divided by 4π: Δx * Δp ≥ h/4π.
• The percentage uncertainty in the momentum of a 1 keV electron whose position is known to within 0.1 nm is approximately 1.05%.

Particle in a Box

• A particle confined to a one-dimensional box of length L has quantized energy levels given by: E_n = (n^2 * h^2) / (8mL^2), where n is an integer representing the energy level.
• The permitted energies of an electron confined in a box 0.1 nm across are quantized according to this equation.

Zero Point Energy

• The zero point energy of a particle in a box is the minimum possible energy it can have, corresponding to the ground state (n=1).
• The zero point energy of an electron confined in an infinite potential well of length 10^-10 m is calculated using the equation above.

Uncertainty in Wavelength

• The relationship between the lifetime of an excited state (τ) and the uncertainty in the energy of the emitted photon (ΔE) is given by: ΔE * τ ≈ ħ, where ħ is Planck's constant divided by 2π.
• The uncertainty in the wavelength of light emitted during a transition from an excited state lasting 10^-3 seconds is related to the uncertainty in energy and the relationship between energy and wavelength.

Compton Scattering

• Compton scattering is the scattering of a photon by a charged particle, typically an electron.
• The de Broglie wavelength of the scattered electron in a Compton experiment depends on the scattering angle and the initial energy of the photon.

Probability in Quantum Mechanics

• The probability of finding a particle in a specific region of space is determined by the square of the magnitude of its wave function, |ψ|^2.
• The probability of finding a particle in the ground state of an infinite square well between x = L/4 and x = 3L/4 can be calculated by integrating the square of the wave function over that region.

Minimum Speed of a Confined Object

• The minimum speed of an object confined to a region of space is determined by the uncertainty principle.
• The minimum speed of a 1 mg object confined between two walls separated by 2 cm can be calculated using the uncertainty principle and the relationship between momentum and speed.

X-ray Scattering

• In Compton scattering, a photon loses energy and its frequency decreases when scattered by an electron.
• The new frequency of an x-ray photon scattered through 90 degrees after colliding with an electron can be calculated using the Compton scattering formula.

De Broglie Wavelength and Kinetic Energy

• The de Broglie wavelength of a particle is inversely proportional to its momentum, which is directly related to its kinetic energy.
• If the kinetic energy of a particle doubles, its de Broglie wavelength is reduced by the square root of 2.

Position and Momentum Uncertainty

• The uncertainty in the position of a particle is inversely related to the uncertainty in its momentum, as determined by the Heisenberg uncertainty principle.
• The fundamental accuracy with which the position of a bullet and an electron can be determined simultaneously with their speed, given a specific uncertainty in speed, can be calculated using the uncertainty principle.

Energy Uncertainty

• The uncertainty in the energy of an excited state of an atom is related to the lifetime of the state by the uncertainty principle.
• The uncertainty in the energy of an excited state with a lifetime of 10^-8 seconds can be calculated.

Velocity Uncertainty and De Broglie Wavelength

• If the uncertainty in the location of a particle is about equal to its de Broglie wavelength, then the uncertainty in its velocity is about equal to one tenth its velocity.

Momentum and Energy of Electron and Photon

• The momentum of an electron and a photon with the same wavelength is inversely proportional to their wavelength.
• The total energy of an electron and a photon with the same wavelength can be calculated by using the relationship between momentum, energy, and wavelength.

Electron in a Box

• An electron confined to a one-dimensional box of length L has quantized energy levels, and transitions between these levels result in the emission or absorption of photons.
• The ground state energy of an electron in a box can be determined from the energy of the photon emitted during a transition from the first excited state to the ground state.
• The wave function of the electron in the third excited state can be sketched according to the wave function solutions for a particle in a box.
• Increasing the length of the box will reduce the energy level spacing of the electron.

Tunneling Current

• Tunneling is a quantum phenomenon where a particle can penetrate a potential barrier even if it does not have enough energy to classically overcome it.
• The tunneling current is influenced by the height and width of the barrier.
• A decrease in the distance between the surface and the STM tip leads to an increase in the tunneling current.

Compton Scattering and Energy Transfer

• In Compton scattering, some of the photon's energy is transferred to the recoiling electron.
• The wavelength of the scattered photon can be calculated using the energy conservation principle and the Compton scattering formula.

Quantum Tunneling Probability

• The probability of a particle tunneling through a potential barrier is influenced by the barrier's height, width, and the particle's energy.
• The probability of an electron tunneling through a barrier can be calculated using quantum tunneling theory.

Beam of Electrons and Tunneling Probability

• The energy required for a certain percentage of electrons to tunnel through a barrier can be calculated.

Wave Function and Probability

• The wave function of a particle describes its quantum state and can be used to calculate the probability of finding the particle in a specific region of space.
• The normalization constant A of a wave function ensures that the total probability of finding the particle in any possible location is 1.

Blackbody Radiation

• A blackbody is an idealized object that absorbs all radiation incident upon it and emits radiation at all frequencies.
• The wavelength at which a blackbody emits the most radiation (λmax) is inversely proportional to its temperature.
• If the temperature of a blackbody cavity is increased so that the rate of emission of spectral radiation is doubled, the wavelength at which it emits the most radiation (λmax) will decrease.

Photon Scattering

• Photon scattering can involve energy transfer between the photon and a scattering particle.
• The angle at which a photon must be scattered by a free electron to lose a certain percentage of its energy can be determined using the Compton scattering formula.

Uncertainty in Wavelength and Position

• The uncertainty in the position of a particle is related to the uncertainty in its momentum, which is related to the uncertainty in its wavelength through the De Broglie relation.
• The simultaneous values for Δx (uncertainty in position) and Δλ (uncertainty in wavelength) can be calculated by applying the Heisenberg uncertainty principle considering the relationship between momentum and wavelength.

Explore the concepts of de Broglie wavelength and Heisenberg's uncertainty principle in this quiz. Understand the relationship between a particle's momentum and its associated wavelength, along with the implications of uncertainty in measurements. Perfect for students studying quantum mechanics.