Quantum Mechanics and Spin-1/2 Particles
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Questions and Answers

What results are obtained when measuring the spin of a spin-1/2 particle along any chosen direction?

  • Any continuous value
  • +~/2 only
  • −~/2 only
  • +~/2 or −~/2 (correct)

What happens when a measurement of spin is made along the z-direction?

  • The measurement collapses the wave function entirely.
  • Results will vary randomly in a continuous manner.
  • It yields a consistent value for subsequent measurements along the same direction. (correct)
  • The result is always ambiguous.

If the spin is known along the z-direction, what is the probability of obtaining the same or opposite spin result when measuring along a perpendicular direction?

  • 100% probability for the same result
  • 50% probability for each of ±~/2 (correct)
  • 0% probability for each direction
  • 25% probability for +~/2 and 75% for −~/2

What is the total number of quantized spin projections available for a spin-n particle?

<p>2n + 1 (A)</p> Signup and view all the answers

What does a measurement along the x-direction imply for subsequent measurements along the z-direction?

<p>They become ambiguous with 50% probability for either result. (A)</p> Signup and view all the answers

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

<p>It is defined with two eigenvalues and eigenvectors. (C)</p> Signup and view all the answers

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

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

What is the nature of quantum mechanics as described in relation to spin measurements?

<p>It is the simplest explanation for behavior at the smallest scales. (B)</p> Signup and view all the answers

What does the equation $E = \hbar \omega$ represent?

<p>Einstein's relation relating energy to angular frequency (A)</p> Signup and view all the answers

In the context of scattering, what does the potential $V(x)$ represent for $x < 0$?

<p>A free particle state with zero potential energy (B)</p> Signup and view all the answers

The relationship $p = \hbar k$ signifies what in wave mechanics?

<p>The de Broglie relation linking momentum to wave number (D)</p> Signup and view all the answers

What does the term $\psi(x, t) = \phi(x) T(t)$ indicate in quantum mechanics?

<p>A combined wave function of spatial and temporal components (A)</p> Signup and view all the answers

What is the physical interpretation of the term $V_0$ in the potential step?

<p>The height of the potential barrier encountered by the particle (A)</p> Signup and view all the answers

When solving the TISE in different regions, what boundary condition must be applied?

<p>The wave function and its derivative must be continuous (A)</p> Signup and view all the answers

In the equation for kinetic energy $E = \frac{p^2}{2m}$, what does $p$ represent?

<p>The particle's momentum (A)</p> Signup and view all the answers

For a plane wave described by $ ext{ψ(x, t)}$, what does the term $ ext{e}^{-iωt}$ signify?

<p>The wave's temporal evolution (C)</p> Signup and view all the answers

What is the probability of measuring the state | ↑z i when the system is in the state | ↑x i?

<p>1/2 (D)</p> Signup and view all the answers

Which expression correctly represents the amplitude for measuring the state | ↑z i from the state | ↑x i?

<p>h↑z | ↑x i = √ (1 + 0)/2 (A)</p> Signup and view all the answers

What is the general formula for the probability of measuring state |ϕi when state |ψi is prepared?

<p>|hϕ|ψi|^2 (C)</p> Signup and view all the answers

In the context of repeated measurements, what happens when one output path of the Stern Gerlach apparatus is blocked?

<p>It selects spins with a chosen orientation along a chosen direction. (D)</p> Signup and view all the answers

What does the expression |h↑z | ↑x i| evaluate to?

<p>1/2 (D)</p> Signup and view all the answers

When a state | ↑x i is expressed in the z-basis, what does the expression | ↑x i = √(1/2)(| ↑z i + | ↓z i) represent?

<p>The state is equally likely to be measured as | ↑z i or | ↓z i. (C)</p> Signup and view all the answers

What is the meaning of the notation h↑z | ↑x i in the context of quantum states?

<p>It represents the measurement of | ↑x i when prepared in the | ↑z i basis. (D)</p> Signup and view all the answers

What does the expression √(h↑z | ↑z i) yield?

<p>1 (C)</p> Signup and view all the answers

What happens when you divide both sides of the equation by φT?

<p>You only have functions of time on one side and functions of position on the other side. (B)</p> Signup and view all the answers

Why must both sides of the equation be equal to the same constant?

<p>Because the right side contains space-dependent functions only. (C), Because the left side contains time-dependent functions only. (D)</p> Signup and view all the answers

What does the constant E represent in the context of the equation?

<p>It signifies the energy of the system. (C)</p> Signup and view all the answers

What is the key result of solving the time-independent Schrödinger equation (TISE)?

<p>It gives solutions to the time-dependent Schrödinger equation (TDSE). (A)</p> Signup and view all the answers

What does the expression ψ(x, t) represent?

<p>It denotes the wavefunction at a specific time and location. (B)</p> Signup and view all the answers

Which statement about the time evolution of the wavefunction is true?

<p>It is given by an exponential function involving energy. (B)</p> Signup and view all the answers

What implication does successfully solving the TISE have on the TDSE?

<p>You can describe the wavefunction for all future and past times. (D)</p> Signup and view all the answers

What condition must be true for expressions involving φ(x) and ψ(x, t) to hold?

<p>They must hold in the absence of measurement. (A)</p> Signup and view all the answers

What does the expression | ↑x ih↑x | + | ↓x ih↓x | represent in quantum mechanics?

<p>A superposition of states with definite spin in the x-direction (D)</p> Signup and view all the answers

What is the result of measuring the spin in the x-direction according to the provided content?

<p>It randomizes the result of the z-direction measurement (D)</p> Signup and view all the answers

What is the significance of the gap between the two possible paths followed by the particle?

<p>It allows for macroscopic separation of quantum states (B)</p> Signup and view all the answers

Which aspect differentiates quantum superposition from classical probabilities as mentioned in the content?

<p>Quantum superpositions have uncertain outcomes even when paths are defined (C)</p> Signup and view all the answers

What role does the Stern-Gerlach apparatus play in the described experiment?

<p>It allows for the separation of particle states based on spin (C)</p> Signup and view all the answers

What happens to the z-direction spin information when x-direction measurement information is erased?

<p>It is preserved despite previous measurements (D)</p> Signup and view all the answers

What is the overall implication of the quantum eraser experiment described?

<p>It demonstrates the complex nature of quantum state measurements (C)</p> Signup and view all the answers

How does measuring along x affect the quantum state originally prepared in z?

<p>It completely collapses the z-state (A)</p> Signup and view all the answers

What is a characteristic of the Schrödinger and Heisenberg pictures?

<p>They are equivalent and yield the same amplitudes. (D)</p> Signup and view all the answers

What does the Heisenberg equation of motion express?

<p>It states the relationship between an operator and the Hamiltonian. (A)</p> Signup and view all the answers

When is an observable quantity A associated with operator  considered conserved?

<p>When the derivative of its expectation value with respect to time equals zero. (C)</p> Signup and view all the answers

What condition must be met for an operator  to be considered conserved in relation to the Hamiltonian Ĥ?

<p>The commutator Â, Ĥ must equal zero. (B)</p> Signup and view all the answers

What does the equality in Eq. 226 represent?

<p>The correlation between the Schrödinger and Heisenberg pictures. (D)</p> Signup and view all the answers

What is implied by the statement about amplitudes being the same in both pictures?

<p>The underlying physics remains consistent regardless of the picture used. (D)</p> Signup and view all the answers

What does the term 'arbitrary choice' in Eq. 228 suggest?

<p>The choice of state vector can vary without altering measurable outcomes. (C)</p> Signup and view all the answers

In the context of quantum mechanics, what is primarily affected by a time-dependent potential?

<p>The explicit time dependence of the Schrödinger operators. (D)</p> Signup and view all the answers

Flashcards

Time-Independent Schrödinger Equation (TISE)

A fundamental equation in quantum mechanics that describes the time-independent behavior of a quantum system.

Time-Dependent Schrödinger Equation (TDSE)

The equation that describes the time evolution of a quantum system.

Wavefunction (ψ)

A mathematical function that describes the state of a quantum system. It gives the probability of finding the system in a particular state.

Energy (E)

A constant that represents the total energy of the system.

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Hamiltonian Operator (Ĥ)

A mathematical operator that represents the energy of the system.

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Separation of Variables

A mathematical technique used to separate the time and spatial dependence of the wavefunction.

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Time-Independent Wavefunction (φ)

The solution to the TISE that gives the spatial distribution of the system.

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Relationship between ψ(x,t) and φ(x)

A key relationship between the TDSE solution and the TISE solution.

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Intrinsic Angular Momentum (Spin)

The property of an object related to its rotation, analogous to the angular momentum of a spinning object.

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Stern-Gerlach Experiment

A fundamental experiment in quantum mechanics where a beam of silver atoms with spin-1/2 is passed through a magnetic field gradient.

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Quantization of Spin

The observation that the spin of a particle can only be measured in discrete values, not a continuous range, as in classical physics.

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Einstein's Relation

The equation that relates the energy of a particle to its angular frequency.

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Spin-1/2 State Vector

A mathematical representation of the state of a spin-1/2 particle, which is a two-dimensional vector with eigenvalues of +ħ/2 or -ħ/2.

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Time-Independent Wavefunction (φ(x))

Describes the spatial distribution of a quantum system and is a solution to the Time-Independent Schrödinger Equation (TISE).

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Spin Observable Matrix

A 2x2 Hermitian matrix that represents a spin observable, corresponding to the measurement of spin along a specific direction.

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Boundary Conditions in Quantum Mechanics

The condition that a wavefunction and its derivative must be continuous at the boundary between two different regions.

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Quantum Weirdness of Spin

The random change in the spin measurement along the z-axis after performing a measurement along the x-axis, even though repeated measurements along z would otherwise give the same result.

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Spin Hilbert Space

The set of all possible spin states that a particle can be in, which is a two-dimensional complex vector space due to the two possible eigenvalues.

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Quantum Tunneling

A concept in quantum mechanics where waves can pass through barriers even if they don't have enough energy to climb over them due to quantum tunneling.

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Quantum Mechanics of the Universe

The explanation of the behavior of the universe at the smallest scales based on the concepts of quantum mechanics, such as quantization and superposition.

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de Broglie Relation

The de Broglie relation states that the momentum of a particle is directly proportional to its wavevector.

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Heisenberg vs. Schrödinger Picture

The Heisenberg picture describes a quantum system where the operators evolve in time, while the states remain constant. In contrast, the Schrödinger picture describes a system where the states evolve in time while the operators remain constant.

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Heisenberg Equation of Motion

In the Heisenberg picture, the time evolution of an operator is described by the Heisenberg equation of motion. This equation tells us how the operator changes with time under the influence of the Hamiltonian.

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Conserved Quantity in Quantum Mechanics

A quantity is conserved if its expectation value remains constant over time. This means that the observable associated with the operator does not change with time.

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Commutation Relation for Conserved Quantities

An observable is conserved if and only if its corresponding operator commutes with the Hamiltonian. This means that the observable's value doesn't change over time.

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Expectation Value in Quantum Mechanics

In quantum mechanics, the expectation value of an observable is the average value of the observable when measured on a system in a given state. It is calculated using the wavefunction of the system.

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Quantum Eraser

The process of removing information about a previous measurement by recombining the possible outcomes, allowing the initial state to remain intact.

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Quantum Eraser Experiment

This experiment highlights the fundamental difference between a quantum superposition and a classical probability. In quantum mechanics, the state of a system is not determined until a measurement is made. In classical physics, the state of a system is already determined, and we are simply observing it.

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Measuring Spin in Different Directions

The process of passing a particle through an intermediate measurement apparatus before measuring its spin in a different direction. Performing an initial measurement along the x-axis randomizes the result of measuring the spin along the z-axis.

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Quantum Superposition vs. Classical Probability

A classical probability describes an event with two known outcomes, while a quantum superposition describes a state with two possible outcomes that are only realized upon measurement.

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Stern-Gerlach Experiment and Eraser

An experiment that illustrates the bizarre behavior of quantum particles, where the information about a previous measurement can be erased, restoring the initial state.

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Probability Amplitude

The probability amplitude for measuring a state |ϕi when a system is in state |ψi is given by the inner product of the two states: hϕ|ψi.

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Probability of Measurement

The probability of measuring a state |ϕi when a system is prepared in state |ψi is calculated by squaring the magnitude of the probability amplitude: |hϕ|ψi|²

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Superposition of Spin States

A |↑x> state, where the spin of the particle is aligned along the x-axis, can be written as a linear combination of the |↑z> and|↓z> states, each with a coefficient of √(1/2). This means the particle has an equal probability of being measured in either the up or down spin state along the z-axis.

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Probability for |↑z> in |↑x> state

The probability of measuring |↑z> when a system is in the |↑x> state is 1/2, indicating an equal chance of finding the spin aligned up or down along the z-axis.

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Stern-Gerlach Apparatus

To measure the spin orientation of a particle along different directions, we can use a series of Stern-Gerlach apparatuses, each oriented along a particular direction.

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Spin Selection

By blocking one output path of a Stern-Gerlach apparatus, we can select only particles with spins oriented in a specific direction.

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Overlap Between States

The probability amplitude hϕ|ψi represents the overlap between the state |ψi and the state |ϕi. It encapsulates the likelihood of transitioning from the initial state |"ψi> to the final state |"ϕi> during a measurement.

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Transition Probability

The probability |hϕ|ψi|² is the square of the probability amplitude, representing the actual likelihood of finding the system in state |"ϕi> after a measurement when prepared in state |"ψi>.

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