Why does Hebrews 1 quote Psalm 102?

Here's a thought I had: comments?

In Psalm 102:25-27, the exile is speaking in faith of his God, that his God is the everlasting, unchanging Creator God; in Hebrews 1:8-12, in which this verse is applied to the Son, the Father is speaking. So what is the writer to the Hebrews doing?
In order to believe in his situation that Zion really can be restored, the exile needs to believe that his God is the unchanging, faithful, everlasting Creator God, fully able and fully faithful. In order to believe that His Messiah can really achieve the mission for which He has appointed Him, that of second Adam and therefore restorer of Eden itself, of Zion, the City of God with men, then God Himself must be able to express similar faith in the Messiah! What the writer to the Hebrews therefore implies is that his readers may have full faith in the Lord Jesus because He is the one of whom God the Father may speak precisely those words of faith that they have been speaking of God since the exile.