What is a unitary transformation and how does it relate to the rotation of a quantum system in the Hilbert space?
A unitary transformation is a fundamental concept in quantum mechanics that describes the evolution of a quantum system in the Hilbert space. It is a linear transformation that preserves the inner product between vectors, ensuring that the norm and the orthogonality of vectors are conserved. In other words, it preserves the probability amplitudes of quantum states, which are essential for the probabilistic nature of quantum mechanics.
Mathematically, a unitary transformation U is represented by a unitary matrix, which is a square matrix U such that its conjugate transpose U† is equal to its inverse. This can be written as U†U = UU† = I, where I is the identity matrix. The unitary matrix U acts on a quantum state vector |ψ⟩, transforming it into a new state vector |ψ'⟩ = U|ψ⟩.
The relationship between unitary transformations and the rotation of a quantum system can be understood by considering the analogy with classical physics. In classical mechanics, rotations are described by orthogonal transformations, which preserve distances and angles. Similarly, in quantum mechanics, unitary transformations play the role of rotations in the Hilbert space, preserving the norm and inner product of quantum states.
To illustrate this concept, let's consider a simple example of a spin-1/2 particle, such as an electron. The Hilbert space for this system is two-dimensional, spanned by the basis states |↑⟩ and |↓⟩, representing the spin-up and spin-down states, respectively. We can represent these states as column vectors:
|↑⟩ = [1, 0]ᵀ
|↓⟩ = [0, 1]ᵀ
Now, let's consider a unitary transformation that corresponds to a rotation of the spin-1/2 particle around the z-axis by an angle θ. This transformation can be represented by the matrix:
U = exp(-iθσ₃/2)
where σ₃ is the Pauli matrix corresponding to the z-component of the spin operator. Applying this transformation to the spin-up state, we have:
U|↑⟩ = exp(-iθσ₃/2) [1, 0]ᵀ
Using the matrix representation of σ₃:
σ₃ = [1, 0; 0, -1]
we can calculate the result of the transformation:
U|↑⟩ = [cos(θ/2), -sin(θ/2); sin(θ/2), cos(θ/2)] [1, 0]ᵀ
= [cos(θ/2), -sin(θ/2)]ᵀ
This represents a new quantum state that corresponds to a superposition of the spin-up and spin-down states, with a relative phase determined by the angle θ. Similarly, applying the unitary transformation to the spin-down state, we obtain:
U|↓⟩ = [cos(θ/2), sin(θ/2)]ᵀ
This demonstrates how a unitary transformation can rotate the quantum state of a spin-1/2 particle in the Hilbert space.
A unitary transformation is a linear transformation that preserves the inner product of quantum states, ensuring the conservation of probability amplitudes. It plays the role of rotations in the Hilbert space, allowing the evolution of quantum systems and the transformation of quantum states. The relationship between unitary transformations and the rotation of a quantum system is evident in the preservation of norm and orthogonality, analogous to the preservation of distances and angles in classical rotations.
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