hi everyone this is professor m science and today i want to talk about unitary operators in another one of our videos on rigorous quantum mechanics unitary operators are used for many different purposes in quantum mechanics for example spatial translations or time evolution are described by unitary operators as another example the symmetry operations of a hamiltonian can be represented by a group of unitary operators the aim of today's video is to go over the basic mathematical properties of unitary operators so we can use them confidently when we need to let's go a unitary operator u is an operator whose inverse is equal to its adjoint and the definition is as simple as that in the rest of the video we'll explore the properties of operators that obey this relation but before we do that let's rewrite the definition in another way that is also typically used remember that the inverse of an operator is such that when we multiply them together in either order we obtain the identity for unitary operators this means that u dagger u is equal to uu dagger which is equal to the identity this equation is a restatement of the fact that u dagger is equal to the inverse of u the first property of unitary operators that we're going to look at is that the product of two unitary operators is also unitary to see this consider a unitary operator u another unitary operator v and the product uv to check that uv is also unitary we first consider uv dagger times uv we know that the adjoint of a product of two operators is equal to the product of the add joints in reverse order and we get v dagger u dagger u v as u is unitary this is the identity and we get v dagger v and as v is unitary this also gives the identity we could similarly show that u v times u v dagger is also equal to the identity and putting these results together we see that uv is unitary this confirms that the product of two unitary operators is itself a unitary operator now i want to consider the eigenvalue equation of unitary operators we can write it as u lambda equals lambda lambda where as usual the lambda here is the eigenvalue and the lambda here the eigenstate we've been working a lot with hermitian operators in our study of quantum mechanics and in that case the eigenvalues are real numbers however for a general non-hermitian operator like the unitary operators we're looking at today the eigenvalues don't have to be real numbers and in general they'll be complex numbers to figure out the eigenvalues let's consider the norm squared of u lambda we can use the eigenvalue equation to write this as the norm squared of lambda lambda we can expand the norm as the bra times the ket remembering to take the complex conjugate of the lambda associated with the bra assuming that the eigenstates are normalized then we end up with the absolute value squared of lambda let's now write the norm squared again we now first expand in terms of the bra and the ket as u is unitary we end up with lambda lambda which equals 1. putting these results together we get that the absolute value squared of lambda is equal to 1. this means that the eigenvalues of a unitary operator are numbers of magnitude 1.
we can write any number of magnitude 1 as this exponential where phi lambda is a real number so the eigenvalues of unitary operators are in general complex numbers but their magnitude must be equal to 1. this is quite different to what we know about the eigenvalues of hermitian operators which can take any value but they must be real numbers still working with the eigenvalue equation the next thing i want to show is at the eigenstate of unitary operator that corresponds to different eigenvalues are orthogonal to do so we start again with the eigenvalue equation of a unitary operator u the first thing we need to show is that this implies that the bra lambda u is equal to the eigenvalue times the bra lambda to see this we start with the eigenstate lambda we then insert the identity we can write out the identity as u dagger u because u is unitary this here is the eigenvalue equation for u so we get lambda u dagger lambda using the result from the previous slide that lambda is a number of magnitude 1 we can rewrite this as the exponential of i phi lambda times u dagger lambda we can then isolate this term and get u dagger lambda equal to the exponential of minus i phi lambda times lambda this is just the complex conjugate of the original exponential so we can rewrite this expression as the complex conjugate of lambda times lambda the final step is to convert this latest equation to the joule space to get this and as usual we got this by turning all cats into brass here and here all scalars into their conjugates here and all operators into their adjoints here this completes the proof of this relation up here and we're now ready to show that the eigenstates are orthogonal let's first make some room we consider two eigen states of u the first lambda a base d eigenvalue equation and the second mu a base this other eigenvalue equation for the second one we'll also make use of the corresponding equation for brass we've just derived okay we're now ready for the proof first we calculate the bracket mu u lambda using this eigenvalue equation in this part here gives lambda mu lambda we can write the bracket again and now we use this expression here in this part here and we end up with mu mu lambda subtracting these two equations gives that zero equals lambda minus mu times the bracket mu lambda this means that if lambda is not equal to mu then this bracket must be zero and that's it the eigenstates of a unitary operator corresponding to distinct eigenvalues are orthogonal up to this point we've investigated the fundamental properties of unitary operators that follow from their definition up here in the rest of the video we'll look into what happens when we apply a unitary operator to a quantum state or to another operator this process of applying a unitary operator is called a unitary transformation a key reason why unitary operators are so important in quantum mechanics is that unitary transformations conserve the scalar product let's consider a ket psi 1 prime equal to the action of a unitary operator u on a ket psi 1. this is an example of a unitary transformation between the state psi 1 and the state psi 1 prime we can do the same with another state si2 prime which is the unitary transformation of the state psi 2.
we now ready to prove that unitary transformations conserve the scalar product to do so let's consider the scalar product between psi 1 prime and site 2 prime we can rewrite it in terms of the unitary transformation of the bra of psi 1 remembering to take the adjoint of u and going to dual space and then the unitary transformation of psi 2. this here is now the identity operator because u is a unitary operator and we get the bracket psi 1 psi 2.
