To show that a unitary operator remains unitary under a unitary transformation, we need to demonstrate that the resulting operator still satisfies the properties of unitarity. Let's consider the steps to prove this:
Unitary Operator: First, let's assume we have a unitary operator U, which means it satisfies the condition U†U = UU† = I, where U† denotes the Hermitian adjoint (conjugate transpose) of U, and I represents the identity operator.
Unitary Transformation: Now, suppose we have a unitary transformation V, which is also a unitary operator. This means V†V = VV† = I.
Applying the Transformation: We want to show that the operator W = VUV† remains unitary after the unitary transformation. To do this, we need to prove that WW† = W†W = I.
Calculating W†W: Let's calculate W†W. First, we take the Hermitian adjoint of W, which gives us (VUV†)† = V†U†V. Then, we multiply this result by W to obtain (V†U†V)(VUV†).
Simplification: Using the properties of the adjoint operation and the fact that V and V† are both unitary operators (V†V = VV† = I), we can simplify the expression to V†U†UV = V†VU†UV.
Unitarity of U: Since U is a unitary operator, we know that U†U = UU† = I. Therefore, we can replace U†U in the expression with I, yielding V†VI = V†V = I.
Final Result: We have now shown that W†W = V†VU†UV = I, which demonstrates that the operator W = VUV† remains unitary under the unitary transformation.
In summary, if we start with a unitary operator U and apply a unitary transformation V, the resulting operator W = VUV† is also unitary. This proof relies on the properties of unitary operators, the Hermitian adjoint, and the identity operator.