Quantum Mechanics: Wavefunction and Probability
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Questions and Answers

What does the Born rule state regarding the wavefunction ψ?

  • The wavefunction is always zero.
  • The wavefunction must be continuous everywhere.
  • Wavefunction values can be negative.
  • The probability to find a particle between two points is related to the wavefunction squared. (correct)

    • What is the significance of the integral of the probability density ρ(x, t) over all space?

  • It indicates the maximum distance the particle can travel.
  • It gives the kinetic energy of the particle.
  • It is always equal to 1. (correct)
  • It represents the velocity of the particle.

    • Which statement correctly describes the probability density ρ(x, t)?

  • It must have a unit of velocity.
  • It is the square of the wavefunction. (correct)
  • It must always be negative.
  • It is equal to the wavefunction itself.

    • What property do infinite potentials have in quantum mechanics?

    <p>Particles avoid regions of infinite potential. (C)</p> Signup and view all the answers

    What do the properties of differential equations indicate about the time-independent Schrödinger equation (TISE)?

    <p>It is a second-order equation in spatial derivatives. (C)</p> Signup and view all the answers

    Which statement is a consequence of the global conservation of probability?

    <p>The overall probability of finding the particle is always 1. (A)</p> Signup and view all the answers

    What does the integral expression Z a^b |ψ|^2 dx represent?

    <p>The probability of finding the particle between a and b. (A)</p> Signup and view all the answers

    Which property of potentials is generally accepted in quantum mechanics?

    <p>Potentials can be infinite or discontinuous but not pathological. (A)</p> Signup and view all the answers

    What is the expression for the probability current density for the incident wave?

    <p>$\frac{\hbar k_I}{m}$ (A)</p> Signup and view all the answers

    Which equation represents the probability of reflection?

    <p>$R = \frac{k_I - k_{II}}{k_I + k_{II}}$ (B)</p> Signup and view all the answers

    What is the expression for the probability of transmission?

    <p>$T = \frac{2k_{II}}{k_I(k_I + k_{II})}$ (D)</p> Signup and view all the answers

    What does the equation $R + T = 1$ represent?

    <p>The conservation of probability (C)</p> Signup and view all the answers

    Which term is used to denote the probability current density for the reflected wave?

    <p>$j_R$ (D)</p> Signup and view all the answers

    In the equation for probability current density, what does the negative sign in $j_R$ indicate?

    <p>It indicates the direction of the vector quantity. (C)</p> Signup and view all the answers

    The probability of transmission is directly proportional to which of the following?

    <p>|t| (D)</p> Signup and view all the answers

    Which of the following formulas represents the probability of reflection based on the amplitudes?

    <p>$R = \frac{|r|^2}{k_I^2}$ (D)</p> Signup and view all the answers

    What phenomenon describes a particle being found on the opposite side of a barrier despite the potential being higher than the particle's energy?

    <p>Quantum tunnelling (D)</p> Signup and view all the answers

    Which of the following is NOT a topic covered regarding bound states?

    <p>Particle decay (B)</p> Signup and view all the answers

    What is the primary characteristic of the infinite potential well in quantum mechanics?

    <p>The potential is infinite in certain regions (A)</p> Signup and view all the answers

    What must be proved about any finite potential well?

    <p>It has at least one bound state. (C)</p> Signup and view all the answers

    Which of the following statements about the energy eigenstates in the infinite potential well is true?

    <p>They are normalized wavefunctions. (B)</p> Signup and view all the answers

    What does normalisation of wavefunctions ensure in quantum mechanics?

    <p>That the wavefunction has a probability of one across all space. (A)</p> Signup and view all the answers

    In the context of the finite potential well, what does the TISE stand for?

    <p>Time-independent Schrödinger Equation (D)</p> Signup and view all the answers

    Which of the following phenomena is NOT related to the physical concept of bound states?

    <p>Particles existing in free space (A)</p> Signup and view all the answers

    What does the wavefunction ψ (x, t) represent in quantum mechanics?

    <p>A complex number assigned to each point in space and time (C)</p> Signup and view all the answers

    Which statement correctly describes the Schrödinger equation?

    <p>It was postulated by Erwin Schrödinger and won him a Nobel Prize. (D)</p> Signup and view all the answers

    What does the symbol ~ represent in the time-dependent Schrödinger equation?

    <p>Planck's constant divided by 2π (B)</p> Signup and view all the answers

    What aspect of waves does the wavefunction ψ provide information on?

    <p>The amplitude and phase of the wave (D)</p> Signup and view all the answers

    What fundamental property does the wavefunction maintain?

    <p>It is subject to linear wave superposition. (A)</p> Signup and view all the answers

    The time-dependent Schrödinger equation involves which variable held constant when describing change?

    <p>The position x (B)</p> Signup and view all the answers

    Which term refers to the results obtained when solving the Schrödinger equation?

    <p>Wavefunctions (D)</p> Signup and view all the answers

    What does the left-hand side of the equation represent in the context of time and position?

    <p>Functions of time and their derivatives (C)</p> Signup and view all the answers

    What is the significance of the time-dependent Schrödinger equation (TDSE) in quantum mechanics?

    <p>It forms the basis for constructing quantum wavefunctions. (D)</p> Signup and view all the answers

    Which statement is true regarding the constant E in the equations?

    <p>E is assumed to be real for now (A)</p> Signup and view all the answers

    What is the time-independent Schrödinger equation (TISE)?

    <p>$Ĥφ (x) = E$ (C)</p> Signup and view all the answers

    What describes the time evolution of the wavefunction according to the content?

    <p>$T(t) = exp(-iEt/~)$ (C)</p> Signup and view all the answers

    What connection do TISE and TDSE have according to the content?

    <p>Solving the TISE gives solutions to the TDSE (C)</p> Signup and view all the answers

    What does the symbol $ψ(x, t)$ refer to?

    <p>Solutions to the TDSE (D)</p> Signup and view all the answers

    What implication is made regarding the wavefunction and measurement?

    <p>Measurement reveals the wavefunction at a particular time (A)</p> Signup and view all the answers

    Why can both sides of the equation be equal only to a constant?

    <p>The left side is independent of time, while the right is independent of space (A)</p> Signup and view all the answers

    What was the primary observation in the Stern-Gerlach experiment?

    <p>Discreet deflections in two distinct directions (A)</p> Signup and view all the answers

    What eigenvalues correspond to the spin state of a spin-1/2 particle?

    <p>+<del>/2 and -</del>/2 (B)</p> Signup and view all the answers

    What happens if a spin measurement is taken along a direction perpendicular to the one previously measured?

    <p>It has a 50% probability for either +<del>/2 or -</del>/2 (A)</p> Signup and view all the answers

    What is the mathematical representation of the spin-1/2 particle's state?

    <p>Two-dimensional complex vector space (C)</p> Signup and view all the answers

    Why were silver atoms chosen for the Stern-Gerlach experiment?

    <p>They are charge neutral and have spin (A)</p> Signup and view all the answers

    What does a repeated measurement of spin along the same direction yield?

    <p>Consistently the same result (D)</p> Signup and view all the answers

    What are the observable quantities associated with spin measurements represented by?

    <p>2 × 2 Hermitian matrices (A)</p> Signup and view all the answers

    What does the observation of quantization in the Stern-Gerlach experiment illustrate?

    <p>The core principles of quantum mechanics (B)</p> Signup and view all the answers

    Flashcards

    Time-Dependent Schrödinger Equation (TDSE)

    The fundamental equation in quantum mechanics describing the time evolution of a quantum system. It relates the time derivative of the wavefunction to the system's Hamiltonian operator.

    Wavefunction (ψ)

    A mathematical function that describes the state of a quantum system. It assigns a complex number to every point in space and time, providing the complete information about the system.

    Hamiltonian operator (Ĥ)

    The energy operator in quantum mechanics. It acts on the wavefunction to determine its corresponding energy.

    Time Derivative of Wavefunction (∂ψ/∂t)

    The derivative of the wavefunction with respect to time, keeping position constant.

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    Reduced Planck Constant (~)

    The reduced Planck constant (h/2π). It appears in many fundamental quantum mechanics equations.

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    Linearity of the Schrödinger Equation

    The superposition principle in quantum mechanics states that any linear combination of solutions to the Schrödinger equation is also a solution.

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    De Broglie Relation (p = ~k)

    The de Broglie relation states that the momentum of a particle is proportional to its wavevector. It establishes the wave-particle duality of matter.

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    Einstein Relation (E = ~ω)

    The Einstein relation states that the energy of a particle is proportional to its angular frequency. It forms the basis of the energy-frequency relationship in quantum mechanics.

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    TISE properties

    The TISE is second-order in spatial derivatives, meaning the wavefunction must be continuous and smooth. This also implies that the potential cannot be too pathological, allowing for infinite potentials and discontinuous potentials, but not worse.

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    Born rule

    The probability of finding a particle between two points (x and x+dx) at a given time (t) is given by the square of the wavefunction's magnitude integrated over that region.

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    Probability density (ρ(x,t))

    The probability density function for position is defined as the square of the absolute value of the wavefunction, representing the probability of finding the particle at a given position.

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    Normalization condition

    The integral of the probability density over all space must equal 1, indicating that the particle exists somewhere in the universe.

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    Probability of finding a particle in a region

    The probability of finding a particle between points a and b is calculated by integrating the squared magnitude of the wavefunction over that region.

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    Conservation of probability

    The probability of finding the particle somewhere in the universe remains constant, regardless of time or other particle interactions.

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    Avoiding infinite potentials

    Quantum particles tend to avoid regions of infinite potential due to the mathematical nature of the Schrödinger equation resulting in a zero probability of finding the particle in such regions.

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    Particle's behavior in quantum mechanics

    A particle's behavior in quantum mechanics is governed by the probability of finding it in a certain region, not by its definite position like in classical mechanics.

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    Probability Current Density

    The probability current density describes the flow of probability in quantum mechanics. It is a vector quantity that indicates the direction and magnitude of the probability flux.

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    Incident Wave Probability Current Density

    The probability current density for an incident wave is proportional to the wave number kI and the square of the amplitude of the incident wave.

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    Reflected Wave Probability Current Density

    The probability current density for the reflected wave is proportional to the wave number kII and the square of the amplitude of the reflected wave, but with a negative sign due to the wave traveling in the opposite direction.

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    Transmitted Wave Probability Current Density

    The probability current density for the transmitted wave is proportional to the wave number kII and the square of the amplitude of the transmitted wave.

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    Probability of Reflection (R)

    The probability of reflection is the ratio of the reflected wave's probability current density to the incident wave's probability current density.

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    Probability of Transmission (T)

    The probability of transmission is the ratio of the transmitted wave's probability current density to the incident wave's probability current density, adjusted for the different wave numbers.

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    Reflection and Transmission Coefficients

    The reflection and transmission coefficients depend on the incident and transmitted wave numbers, which are related to the particle's energy and the potential barrier's height.

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    Spin

    The intrinsic angular momentum of a particle, quantized in units of ħ/2 and having two possible values: +ħ/2 and -ħ/2.

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    Stern-Gerlach Experiment

    An experiment where a beam of silver atoms is passed through an inhomogeneous magnetic field, demonstrating the quantization of spin.

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    Spin Quantization

    The measurement of a particle's spin along a specific direction can only yield one of two discrete values, +ħ/2 or -ħ/2.

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    Spin Measurement: Same Direction

    If the spin of a particle is measured along one axis and then measured again along the same axis, the result will be the same.

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    Spin Measurement: Perpendicular Direction

    If the spin is known along one direction (e.g., z), a measurement along a perpendicular direction (e.g., x) is completely unpredictable, with equal probability for either +ħ/2 or -ħ/2.

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    Spin Measurement: Uncertainty

    Even if the spin is measured along the same axis multiple times with consistent results, measuring it along another axis and then back to the original axis will introduce uncertainty in the original measurement.

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    Spin State Vector

    The state of a spin-1/2 particle is represented by a vector in a two-dimensional complex vector space.

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    Spin Observable Matrices

    Spin observables (quantities that can be measured) are represented by 2x2 Hermitian matrices with eigenvalues corresponding to the spin values.

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    Quantum Tunneling

    A quantum phenomenon where a particle penetrates a potential barrier even if its energy is less than the barrier's height. This is possible due to the wave nature of particles, allowing them to 'tunnel' through the barrier with a certain probability.

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    Infinite Potential Well

    A type of potential where the particle is confined to a specific region, usually a line or box, and cannot exist outside of it. The potential is infinitely high outside the region, ensuring the particle remains trapped.

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    Energy Eigenfunction

    A mathematical solution to the time-independent Schrödinger equation (TISE) that describes the allowed energy states of a system. Each eigenfunction corresponds to a specific energy level.

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    Energy Eigenvalue

    The value of energy that a particle can have in a given potential. These energies are quantized, meaning they can only take on discrete values.

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    Normalization of Wavefunctions

    A process that ensures the probability of finding the particle somewhere in space is always equal to 1. It involves scaling the wavefunction to satisfy this condition.

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    Finite Potential Well

    A type of potential that is finite (has a specific value) within a certain region and drops to zero outside. This allows for the possibility of the particle escaping the potential, making it more realistic than the infinite potential well.

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    Bound State

    A state where the particle remains bound within the potential, meaning it doesn't have enough energy to escape. This is a common situation in atoms and molecules.

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    TDSE: What is it?

    The time evolution of a quantum system is described by theTDSE. It establishes a relation between the time derivative of the wavefunction and the system's Hamiltonian operator.

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    E: Energy Constant

    A constant which represents the total energy of the system in the TISE.

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    TISE: Time-Independent Schrödinger Equation

    A fundamental equation in quantum mechanics that gives the relationship between the energy of a system and its wavefunction. It is a time-independent version of the Schrödinger equation.

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    ψ(x, t) = ψ(x, 0) exp(−iEt/~)

    The wavefunction at time t is determined by the wavefunction at time 0 and the energy of the system. It describes how the wavefunction changes with time.

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    Probability of Finding a Particle

    The probability of finding a particle between points a and b is calculated by integrating the squared magnitude of the wavefunction over that region.

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    Study Notes

    Quantum Mechanics Study Notes

    • Quantum mechanics is a theoretical framework that describes the physical properties of nature at the scale of atoms and subatomic particles.
    • It differs significantly from classical physics, which describes the physical world at larger scales.
    • Quantum mechanics is fundamentally probabilistic, meaning that it is not possible to predict with certainty the outcome of a measurement; instead, probabilities are assigned to various possible outcomes.
    • The theoretical framework is built upon postulates and mathematical structures that are essential to model complex quantum phenomena.

    Key Concepts

    • Schrödinger equation: A linear partial differential equation that describes the evolution of a quantum system over time. It quantifies the energy of the system and its state.
    • Wavefunction: A mathematical function that describes the quantum state of a particle or system. Probability density is related to the square of the absolute value of the wavefunction.
    • Probability density: The probability of finding a particle at a given point in space or momentum. It is proportional to the square of the absolute value of the wavefunction.
    • Probability current: A vector-valued function that describes the probability flow of a quantum system. It represents how the probability density changes over time.
    • Operators: Mathematical objects that act on wavefunctions and related constructs (vectors or functions) to change the system's state. Examples include the Hamiltonian, momentum, and position operators.
    • Eigenvalues and eigenstates: When an operator acts on a particular state of the system, the resulting state changes but it retains proportion to the original. Eigenvalues and eigenstates are those that show up in the calculated results.
    • Quantum numbers: Properties that describe the state of quantum particles, such as energy and angular momentum. These properties are often quantized, i.e. they can only take on discrete values.
    • Superposition: A quantum system can exist in multiple states simultaneously. This is described mathematically as a superposition of different states.
    • Quantum superposition: A quantum system can be in a combination of multiple states simultaneously. This concept is fundamental to quantum mechanics.
    • Measurement problem: There are different viewpoints and philosophies about the interpretation of quantum mechanics, some concerning how measurement outcomes affect the existing wavefunction.

    Important Equations

    • Time-Dependent Schrödinger Equation (TDSE): iħ∂ψ/∂t = Ĥψ
    • Time-Independent Schrödinger Equation (TISE): Ĥψ = Eψ

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    PX2132 Introductory Quantum Mechanics Notes (PDF)

    Description

    This quiz explores key concepts of quantum mechanics related to the wavefunction and probability densities. You'll delve into the Born rule, the significance of the probability density integral, and the properties of potentials in quantum systems. Test your understanding of these fundamental principles.

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