You need to replenish the nutrition that you're body has just spent in performing the workout AND supply it with fuel for the muscle and
connective tissue recovery that will follow. If you don't eat, then you'll simply be wasting your workout—you will have made your muscles severely thirsty for nutrition, then simply denied it.
My guess is that if you have felt sluggish after exercising in the morning, then it has been because there's been something lacking in your post-workout nutrition plan. If you eat, for example, too many simple carbs in a single sitting, then your insulin production will climb excessively ... resulting in a crash and physical fatigue. On the other hand, if you refuse nutrition to your body after working out in the morning, then your body is not only lacking the fuel/energy to recooperate, it has no fuel for your post-workout activities (what you're doing for the rest of the day).
If you have a choice, workout in the morning, then properly fuel your body. As you continue through your day, your metabolism will be kicked into gear and your muscle structure will be growing stronger by the minute as your
lean tissue repairs itself from your workout efforts. Going to sleep right after a workout is less than optimal—you've just stepped up your metabolism ... going to sleep will simply kill the metabolic benefits.