Let me begin by repeating that I would have voted against QE3. That said, Ben Bernanke is a very good economist with honorable intentions, so there's some good reasons for it. Here's the case for QE3.
The key aspect of QE3 involve the intent to purchase $40 billion worth of mortgage-backed securities as long as the unemployment rate remains high and even as the economy begins to strengthen. That tells us a few points behind the rationale. Of course the weakness of the recovery is reason for further stimulus, but the design of QE3 suggests a significant concern about the long-run behavior of the labor market. Ben Bernanke and others on the Fed fear hysteresis. What's that? It occurs when a significant number of the unemployed remain out of work for an extended period of time, rendering them unemployable. Thus, short-term unemployment resulting from a recession results in long-term unemployment which becomes a permanent fixture of the economy. That could mean they leave the labor force and don't come back (witness the significant decline in the labor force paritcipation rate in recent years) or they remain in the labor force and become structurally unemployed (want a job but aren't qualified for the jobs that are available). As a resuslt, the "normal" unemployment rate (what economists call the natural rate or NAIRU) rises. A consensus seems to be developing that the "normal" unemployment rate has risen from just below 5% to 6% (still some diagreement with some thinking it could be as high as 7% while others think it is somewhat lower; hard to be sure given it's still changing). What's the big deal about an increase of about one percentage point? One percent translates to about 1.5 million people (given the current labor force of about 155 milion). So the human cost is quite high. In addition, with 1.5 million fewer workers employed during normal times, the economy won't be able to produce as much, reducing potential GDP. For now, let's assume that these permanently unemployed workers that remain in the labor force are half as productive as the average worker. Taking GDP divided by the total number of people employed gives the average output per worker ($109,000). Divide that in half to estimate output per worker becoming permanently unemploed. Multiply that by 1.5 million and it's over $80 billion per year (not including those that permanently drop out of the labor force). In order to reduce the likelihood of this permanent increase in unemployment and loss of output, the Fed thinks it's worth doing it all can to reduce unemployment and thus reduce the likelihood of hysteresis.
Why mortgage-backed securities instead of US Treasuries? One reason for the very weak recovery is that residential investment is not rebounding like in previous recoveries. Following the 1973-75 and 1982 recessions, residential investment contributed an average of about one percentage point to economic growth for the first two years of recovery. What about this time? In 2010 and 2011, residential investment's contribution was slightly negative. Also, some of the highest unemployment rates are in occupations related to construction, so targeting mortgages instead of interest rates in general can have a more significant effect on unemployment. In addition, when people buy houses, they also tend to also buy furniture as well as other purchases, so some may say that spurring the housing market could help with sales of furniture and other durable household equipment (the government's term, not mine). In the recoveries of the mid-1970s and early-1980s, this category contrubuted about 0.2% to economic growth for the first two years compared to about 0.1% this time (not much of a difference).
One other change was the statement that the Fed would continue to engage in QE3 even after signs of economic strengthening. Previously, it was thought that the Fed would back off when positive economic data started to take hold. Michael Woodford, an economist with the St. Louis Fed, presented a paper at a central bank meeting in Jackson Hole critical of this policy. He made the case that this sends the signal that the Fed was implying that it was going to provide stimulus because the economy was quite weak, which may reduce consumer and business confidence. By stating that it would continue to provide stimulus even after there are signs of economic strengthening, it may be able to provide stimulus without damaging confidence, resulting in a more powerful effect.
Summing up, inflatio is under control (below 2%), unemployment remains stubbornly high with signs of a permanent increase in the normal rate of unemployment, resulting in singificant human costs and permanent loss of output. Extraordinary efforts to reduce unemployment can thus provide long-term benefits to the economy.