The Sins of Another

One thing I don't understand about the Atonement is summed up fairly clearly in Alma 34:11.

Now there is not any man that can sacrifice his own blood which will atone for the sins of another. Now, if a man murdereth, behold will our law, which is just, take the life of his brother? I say unto you, Nay.

Yet, God's law, which is more just than any earthly law, takes the blood of our spiritual brother, Jesus Christ, to atone for our sins. How could that possibly work? Amulek goes on to say that this is possible because the Savior's sacrifice was an infinite and eternal sacrifice, but I still don't get it. Even an infinite sacrifice from one person wouldn't satisfy the demands of justice from another. If I commit a crime, I have to serve the sentence, no matter who else is willing to do it for me.

Yet, we know that the Atonement works somehow, and thank God that it does! I may not be able to wrap my head around it, but I know that the Atonement is a vital, redeeming force in everyone's lives, especially mine. I have been richly blessed by the miraculous, vicarious power of the Atonement. I don't know how it works, but I know that it does. I suppose I shouldn't look a gift horse in the mouth. I don't need to be smart enough to understand how the Atonement satisfies the demands of justice. I just need to be wise enough to use it.