Quantum Mechanics Basis Sets Quiz

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What remains unchanged when using different basis sets to express a state?

The mathematical operations performed

The magnitude and direction of the electric field (correct)

The specific coordinates used

The direction of the state

Using different basis vectors allows for expressing the same quantum state in multiple ways.

True

What is the outcome when a magnetic moment is placed in a uniform magnetic field?

It precesses about the direction of the field.

The quantum state |VO can be expressed as a superposition of the states |+x) and |—x) multiplied by their respective ________.

amplitudes Signup and view all the answers

Match the following terms with their descriptions:

Basis Set = A set of vectors used to describe a space Column Vector = A representation of quantum states in a specific basis Superposition = A combination of two or more states Electric Field = A physical field surrounding electric charges Signup and view all the answers

Which of the following describes how we might express the electric field in another coordinate system?

By changing the unit vectors involved Signup and view all the answers

Choosing a different basis vector affects the state being represented.

False Signup and view all the answers

The column vector representing the ket |+x) reflects the amplitudes in the ________ basis.

Sx Signup and view all the answers

What is the overall phase that is generally chosen in the provided context?

0 Signup and view all the answers

The state with $S_v = rac{h}{2}$ can be represented as $(2.17a)$ in the S basis.

True Signup and view all the answers

What does the bra corresponding to the ket $|+y angle$ represent in the given basis?

$rac{1}{ ext{constant}}(|-, -o angle)$ Signup and view all the answers

In the equation for ket representation, $|+y angle = rac{1}{ ext{________}}(|+z angle + i|-z angle)$

$ ext{sqrt{2}}$ Signup and view all the answers

Match the following elements to their corresponding representations:

$| -x angle$ = Normalized state with phase $0$ $| +y angle$ = Sum of states in S basis $| +z angle$ = State representation without phase Bra vector$ = Corresponds to the ket vector with $-i$ Signup and view all the answers

What factor appears in the representation of the bra vector for ket $| +y angle$?

-i Signup and view all the answers

The appearance of a complex number in the bra representation is for normalization purposes.

False Signup and view all the answers

State the equation represented in $(2.15)$ for $| -x angle$.

$rac{1}{ ext{constant}}(+y)$ Signup and view all the answers

What is the result of applying the rotation operator to the state |+x)?

|+y> Signup and view all the answers

What is the form of the eigenvalue equation represented as?

-z|A|-z = (+z| Signup and view all the answers

The eigenvalues for the states |+z> and |-z> are the same.

False Signup and view all the answers

What is the significance of the factor of /h in the defining relation?

It relates the infinitesimal rotation operator to the generator of rotations. Signup and view all the answers

The matrix representation of the operator A is a 3 x 3 matrix.

False Signup and view all the answers

What does the matrix element Aij represent?

The action of operator A on basis states labeled by i and j. Signup and view all the answers

The operator that generates rotations about the z axis acts on the spin-up-along-z state, resulting in ______ times the state (the eigenstate).

a constant Signup and view all the answers

The action of the projection operator P+ on the basis states is given by ______.

c:)C)=o Signup and view all the answers

Match the states with their corresponding spin orientation:

|+z> = Spin up along z |-z> = Spin down along z |+x> = Spin up along x |+y> = Spin up along y Signup and view all the answers

Match the following operators with their representations:

P+ = Matrix representation includes < + z | P+ | + z > P_ = Matrix representation includes < - z | P_ | - z > I = Identity matrix Jz = Generator of rotations about the z axis Signup and view all the answers

After rotating the state |+x> by 90°, which mathematical expression describes the resulting state?

e~i7t/4|+y> Signup and view all the answers

The application of a rotation operator produces the same state if the states differ by an overall phase.

True Signup and view all the answers

What does the completeness relation P+ + P_ equal in matrix form?

I Signup and view all the answers

What happens to the state |+z> when the rotation operator is applied?

The resulting state is influenced by the z component of intrinsic spin angular momentum. Signup and view all the answers

The notation (i|A|j) signifies the matrix elements of the operator A.

True Signup and view all the answers

What is the significance of expressing operators in matrix form?

It allows for the easy computation of operator actions on states. Signup and view all the answers

What condition must be satisfied for $J_z |+z\rangle$ to equal a constant times $|+z\rangle$?

$J_z |+z\rangle = c |+z\rangle$ Signup and view all the answers

Two states that differ only by an overall phase are considered to be the same state.

True Signup and view all the answers

What do we call a state that satisfies the eigenstate condition for an operator?

Eigenstate Signup and view all the answers

In the Stern-Gerlach experiments, the eigenvalues for the spin-up and spin-down states are observed to be represented by _____ in the equation given.

±ħ/2 Signup and view all the answers

Match the following states with their corresponding spin eigenvalues:

|+z⟩ = ħ/2 |-z⟩ = -ħ/2 |+x⟩ = Unknown |-x⟩ = Unknown Signup and view all the answers

What results might occur if the condition $J_z |+z⟩$ is not met?

The application of $R(0k) |+z⟩$ will yield $|+z⟩$ along with a term in $|+x⟩$. Signup and view all the answers

The terms in the Taylor series expansion of the rotation operator can cancel out unwanted terms.

False Signup and view all the answers

What is the significance of an operator resulting in a constant times a state in quantum mechanics?

It indicates that the state is an eigenstate of the operator. Signup and view all the answers

What is the result of the adjoint operator A acting on the ket |V0?

A|V0> = I Signup and view all the answers

The state |+x) and |-x) are related to |+z) and |-z) through a rotation operator.

True Signup and view all the answers

What transformation allows the transition from the Sz basis to the Sx basis?

Insertion of the identity operator between the S2 matrix elements. Signup and view all the answers

The state |+x) can be expressed as |+z> + |-z).

1/sqrt(2), 1/sqrt(2) Signup and view all the answers

Match the following states with their corresponding expressions:

|+x> = 1/sqrt(2)(|+z> + |-z>) |-x> = 1/sqrt(2)(|+z> - |-z>) |+z> = Standard basis state |-z> = Standard basis state Signup and view all the answers

Which of the following equations demonstrates the relationship between |±x> and |±z>?

|±x> = R|±z> Signup and view all the answers

The expectation value (Sz) for the state |+x) can only be evaluated in the Sz basis.

False Signup and view all the answers

The expectation value of Sz for the state |+x> has a matrix form represented as _____.

(2.108) Signup and view all the answers

Quantum Mechanics Basis Sets Quiz

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Questions and Answers

What remains unchanged when using different basis sets to express a state?

Using different basis vectors allows for expressing the same quantum state in multiple ways.

What is the outcome when a magnetic moment is placed in a uniform magnetic field?

The quantum state |VO can be expressed as a superposition of the states |+x) and |—x) multiplied by their respective ________.

Match the following terms with their descriptions:

Which of the following describes how we might express the electric field in another coordinate system?

Choosing a different basis vector affects the state being represented.

The column vector representing the ket |+x) reflects the amplitudes in the ________ basis.

What is the overall phase that is generally chosen in the provided context?

The state with $S_v = rac{h}{2}$ can be represented as $(2.17a)$ in the S basis.

What does the bra corresponding to the ket $|+y angle$ represent in the given basis?

In the equation for ket representation, $|+y angle = rac{1}{ ext{________}}(|+z angle + i|-z angle)$

Match the following elements to their corresponding representations:

What factor appears in the representation of the bra vector for ket $| +y angle$?

The appearance of a complex number in the bra representation is for normalization purposes.

State the equation represented in $(2.15)$ for $| -x angle$.

What is the result of applying the rotation operator to the state |+x)?

What is the form of the eigenvalue equation represented as?

The eigenvalues for the states |+z> and |-z> are the same.

What is the significance of the factor of /h in the defining relation?

The matrix representation of the operator A is a 3 x 3 matrix.

What does the matrix element Aij represent?

The operator that generates rotations about the z axis acts on the spin-up-along-z state, resulting in ______ times the state (the eigenstate).

The action of the projection operator P+ on the basis states is given by ______.

Match the states with their corresponding spin orientation:

Match the following operators with their representations:

After rotating the state |+x> by 90°, which mathematical expression describes the resulting state?

The application of a rotation operator produces the same state if the states differ by an overall phase.

What does the completeness relation P+ + P_ equal in matrix form?

What happens to the state |+z> when the rotation operator is applied?

The notation (i|A|j) signifies the matrix elements of the operator A.

What is the significance of expressing operators in matrix form?

What condition must be satisfied for $J_z |+z\rangle$ to equal a constant times $|+z\rangle$?

Two states that differ only by an overall phase are considered to be the same state.

What do we call a state that satisfies the eigenstate condition for an operator?

In the Stern-Gerlach experiments, the eigenvalues for the spin-up and spin-down states are observed to be represented by _____ in the equation given.

Match the following states with their corresponding spin eigenvalues:

What results might occur if the condition $J_z |+z⟩$ is not met?

The terms in the Taylor series expansion of the rotation operator can cancel out unwanted terms.

What is the significance of an operator resulting in a constant times a state in quantum mechanics?

What is the result of the adjoint operator A acting on the ket |V0?

The state |+x) and |-x) are related to |+z) and |-z) through a rotation operator.

What transformation allows the transition from the Sz basis to the Sx basis?

The state |+x) can be expressed as ____ |+z> + ____ |-z).

Match the following states with their corresponding expressions:

Which of the following equations demonstrates the relationship between |±x> and |±z>?

The expectation value (Sz) for the state |+x) can only be evaluated in the Sz basis.

The expectation value of Sz for the state |+x> has a matrix form represented as _____.

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The state |+x) can be expressed as |+z> + |-z).