It's common for piano students to tense a hand that is not actively engaged in playing during measured rests.
Beethoven's "Fur Elise," an aspirational piece for so many, is the perfect representation of interactive, woven hands, that flow across from Left to Right, with a spacious margin of relaxed breaths. (as rests are notated) This over-all legato line mosaic that permeates the opening section, should be responsive to an uninterrupted outpouring without intrusive tension in the hands, wrists and arms at any point.
In beautifully phrased music-making, a basic underlying, hand-to-hand motion plays out simultaneously in the present and in the future. Therefore, if one hand stiffens while the other is sculpting a portion of the phrase, then extraneous energy is expended to the sacrifice of a well-shaped, continuous line. (In the outflow of "Fur Elise," in particular, while one hand is not playing, it should gracefully move to its next destination.)
In the following lesson-in-progress snippet, an adult student exerted what was energy-draining in a perceived left hand tightening in Beethoven's character piece.
In this second lesson sample, a youngster, having studied for 4 and 1/2 months, plays a duet with me with a nice interaction of her hands in relaxed motion. Having been trained from the start with the image of "weeping-willow arms" and supple wrists, she's well imbued with an approach that will further her progress.
Many students, often unconsciously, tense a hand that is not playing in synchrony with the other. By reinforcing the hanging hands off relaxed arms framing, and replaying videos of what needs amending, pupils will practice relaxation techniques that free both hands of tension.
Original Content: Piano Technique: Energy-saving, Relaxed, Resting hands