Quantum Mechanics Basis Sets Quiz

Ensuring that the probability of finding a particle in all possible states sums to 1. This is achieved by multiplying the ket by a normalization factor, often involving square roots.

The overall phase factor (θ) in a ket is typically chosen to be 0, making the ket simpler. This is generally the convention used unless otherwise indicated.

The ket |—x) in the Sz basis is written as (2.12) where the factor √2 is chosen for normalization. This means that the particle has a probability of 1/2 to have spin up and 1/2 to have spin down.

The state |+y) can be expressed in the S basis as a linear combination of the basis states |+z) and |-z). This representation is important for understanding the relationship between different spin states.

The bra vector corresponding to the ket |+y) is denoted as (+y| and is represented in the S basis by (2.17b). Note the appearance of the —i in the representation for the bra vector.

The inner product of a bra and a ket, denoted as (+y|/O—x), represents the probability amplitude of finding a particle in state |—x) when it is actually in state |+y) . This can be calculated using the bra and ket representations in the appropriate basis.

In some cases, it might be advantageous to choose the global phase θ to be π/2 (or 90 degrees). This can be useful for simplifying certain calculations or for emphasizing certain properties of the system.

Using kets and bras provides a more compact and efficient method for deriving quantities like the inner product (e.g. (2.8)). This approach emphasizes the underlying mathematical structure and facilitates calculations.

A set of orthogonal vectors that span a vector space; used to represent a state, such as a quantum state, using linear combinations.

Expressing a quantum state in terms of a specific basis set. This representation determines how the state's properties are measured and visualized.

A quantum state can be a combination of multiple basis states, each with its own amplitude. The amplitudes tell how likely it is to measure the state in each basis state.

A mathematical representation of a quantum state using a column vector, where each element is the amplitude of the state in a particular basis state.

Changing the representation of a state from one basis set to another without changing the state itself.

A mathematical operation that changes the representation of a state by rotating the basis set. In quantum mechanics, this corresponds to rotating the reference frame in which the state is described.

The gradual change in the orientation of a magnetic moment in a magnetic field. In quantum mechanics, this is analogous to rotating a spin state.

The probability amplitude of a basis state in a superposition. The square of its absolute value gives the probability of measuring the state in that particular basis state.

An operator that describes an infinitesimally small rotation about a specific axis. It's related to the generator of rotations by a factor of ħ.

An operator that determines how a system changes when rotated. It's closely tied to the infinitesimal rotation operator.

The process of applying a rotation operator to a quantum state to describe how it changes after a rotation.

A constant value that is obtained when an operator acts on a specific eigenstate.

A state that remains unchanged (except for a constant factor) when acted upon by a particular operator.

An intrinsic property of a particle that describes its angular momentum even when it's not rotating.

The specific component of the spin angular momentum of a particle that is measured along the z-axis.

A particle with only two possible spin states, spin up and spin down, where each state is quantized and represented by the spin angular momentum of ±ħ/2.

The matrix representation of an operator A acting on a ket |V> which satisfies A |V> = |V> (Equation 2.89) is called the adjoint operator matrix.

Rotating the state |-z) by 90 degrees around the y-axis results in the state |—x) defined as (2.90).

The state |+x) is obtained by applying a rotation operator (R) to the state |+z) according to Equation (2.91a).

The matrix representation of the spin operator Sz in the Sx basis is given by Equation (2.107).

Inserting the identity operator before and after a matrix in one basis allows us to transform it to another basis using the corresponding transformation matrix (like the S-matrix).

The expectation value of Sz for the state |+x) in the Sx basis can be calculated using the matrix representation of Sz in the Sx basis and the column vector representation of |+x) in the Sx basis, as shown in Equation (2.108).

The eigenvalue equation can be represented as a form where a ket is multiplied by an operator, resulting in another ket multiplied by a scalar. This form highlights the connection between operators and their corresponding eigenvalues.

A ket representing a state can be represented as a column vector in the Sz basis. This vector's components are the amplitudes of the state in the |+z) and |-z) basis states. The Sz basis is a convenient way to represent states with respect to spin up and spin down.

An operator can be represented as a 2x2 matrix in the Sz basis. This matrix represents the operator's action on the |+z) and |-z) states, transforming them into linear combinations of those states.

The elements of a matrix representing an operator in a particular basis can be expressed as inner products. The element Aij corresponds to the inner product (i|A|j), showing the effect of the operator on the state j and its projection onto the state i.

Projection operators, like P+, which project a state onto a specific subspace, can be represented as matrices. The matrix elements of the operator are calculated using the state it projects onto and the basis states.

The completeness relation, which states that the sum of all projection operators equals the identity operator, can be expressed in matrix form. This means that the sum of the projection operator matrices equals the identity matrix.

Projection operators have specific actions on the basis states. P+ projects |+z) to |+z) and P+ projects |-z) to the zero vector (0), illustrating their filtering behavior.

The operator Jz, which generates rotations around the z-axis, can be represented as a matrix in the Sz basis. This matrix is used to calculate the effect of rotations on states in terms of their spin components.

A state that, when acted upon by an operator, results in a scalar multiple of the original state.

The scalar constant that multiplies an eigenstate when acted upon by an operator.

An operator that rotates a quantum state around an axis k by an angle θ. It can be represented as the exponential of the angular momentum operator Jk.

The eigenstate condition for the state |+z) with respect to the angular momentum operator Jz. This means the spin-up state along the z-axis remains unchanged (up to a constant factor) when acted upon by Jz.

Equation describing the eigenvalues of the spin-up and spin-down states for the z-component of angular momentum (Sz). This aligns with the observed values in the Stern-Gerlach experiments.

If Jz |+z) were not a constant multiple of |+z), applying the rotation operator R(θk) would introduce terms with different spin orientations, changing the state |+z). This contradicts the observation that the state only changes by a phase.

The eigenvalue is (h/2) for the spin-up state |+z) and -(h/2) for the spin-down state |-z). These values align with the measured spin projections in Stern-Gerlach experiments.

The Stern-Gerlach experiment physically demonstrates that the spin-up and spin-down states have specific, quantized values for Sz, which are the eigenvalues corresponding to these states. This confirms the eigenstate condition and the quantized nature of angular momentum.

The eigenstate condition indicates that the state only undergoes a phase change under rotation, meaning it does not change its direction with respect to the rotation axis. It preserves its spin orientation.

If |+z) is not an eigenstate of Jz, then the Taylor series expansion of the rotation operator R(θk) acting on |+z) would introduce terms involving other spin orientations (e.g., |+x), violating the condition that the state should only change by a phase during rotation.