A 2012 paper by investigators from Northwestern University in the Journal of Experimental Social Psychology sheds some new light on the latter issue.
Rather than summarizing the study myself, I will quote the excellent New York Times article about the three experiments that were done [emphasis added by me]:
In the first experiment, 58 undergraduates were randomly assigned to wear a white lab coat or street clothes. Then they were given a test for selective attention based on their ability to notice incongruities, as when the word “red” appears in the color green. Those who wore the white lab coats made about half as many errors on incongruent trials as those who wore regular clothes.
In the second, 74 students were randomly assigned to one of three options: wearing a doctor’s coat, wearing an [artistic] painter’s coat or seeing a doctor’s coat. Then they were given a test for sustained attention. They had to look at two very similar pictures side by side on a screen and spot four minor differences, writing them down as quickly as possible.
Those who wore the doctor’s coat, which was identical to the painter’s coat, found more differences. They had acquired heightened attention. Those who wore the painter’s coat or were primed with merely seeing the doctor’s coat found fewer differences between the images.
The third experiment explored this priming effect more thoroughly. Does simply seeing a physical item, like the coat, affect behavior? Students either wore a doctor’s coat or a painter’s coat, or were told to notice a doctor’s lab coat displayed on the desk in front of them for a long period of time. All three groups wrote essays about their thoughts on the coats. Then they were tested for sustained attention. Again, the group that wore the doctor’s coat showed the greatest improvement in attention.
If you wear a white coat that you believe belongs to a doctor, your ability to pay attention increases sharply. But if you wear the same white coat believing it belongs to a painter, you will show no such improvement.
That was the lede in the Times piece which included a discussion of some previous papers on the subject of embodied cognition [defined by the Stanford Encyclopedia of Philosophy as "when aspects of the agent's body beyond the brain play a significant causal or physically constitutive role in cognitive processing."]
The Times article mentioned another example of embodied cognition relevant to those who work in hospitals—people who carry heavy clipboards feel more important.
So embodied cognition is not always a good thing.
However a white coat is not a clipboard. This research tells us that simply wearing a white coat may help us focus and sustain our attention.