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.</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.</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.</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.</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.</p> Signup and view all the answers

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

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

    Which equation represents the probability of reflection?

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

    What is the expression for the probability of transmission?

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

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

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

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

    <p>$j_R$</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.</p> Signup and view all the answers

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

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

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

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

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

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

    What must be proved about any finite potential well?

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

    What fundamental property does the wavefunction maintain?

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

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

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

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

    <p>Wavefunctions</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</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.</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</p> Signup and view all the answers

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

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

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

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

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

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

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

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

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

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

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

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

    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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