Somebody wasn't thinking about residents when your condo was designed!

I would start by looking into whether the noise can be stopped at its origin. I don't know anything about the control panel you mention, but maybe the component generating the hum (big transformer? contactor?) could be isolated with resilient sound control mounts.

If you can't get your condo association to go for that, you'll have to block the sound on your side of the wall. Your best bet would be to bring in an acoustical engineer.

I'm not one of those, but I've worked in audio production studios for many years, and I've watched studios being built. So I'll throw out a few suggestions based on what I've observed.

My understanding is that foam and other soft surface materials are best used to absorb and control noise that's already in your room. For that I'd recommend fabric-covered fiberglass acoustic panels. Some of them look quite nice, if a little cubicle-y.

However, I think that what you really need here is not absorption, but a barrier of sound deadening material.

I'd start by caulking gaps and openings in the existing wall with acoustic caulk. Don't forget to remove receptacle and light switch cover plates, and squirt the caulk into any gaps around them.

From there you can go in several different ways. A quick and easy approach is to cover the wall with mass loaded vinyl (MLV). You can buy it on rolls, usually 4 feet or a little more wide. Typically it's stapled to furring strips. Warning -- don't cheap out on staples. This stuff can be pretty heavy.

Normally the MLV is then covered with another layer of drywall. I saw one home studio where the owner had hung big, heavy, lined floor-to-ceiling drapes in front of it instead. It looked nice and seemed to work well. I'm not so sure that would be building code compliant, though.

Another quickish fix is to use "green glue" (a resilient damping compound that comes in caulking gun tubes) on the back of the new drywall, instead of hanging MLV This helps to isolate the new drywall layer, but the green glue doesn't have MLV's density and sound blocking ability.

The ultimate sound control solution (that I can think of) would be to build a second complete studwall in front of the existing wall and not attached to it. The idea is to leave a dead space a few inches wide between the existing wall and your new one, then add sound absorbing panels between the layers.

Again, I'm not an expert, just suggesting soundproofing methods that I've seen used in studio construction. Others with their own experiences and observations may have further or better ideas for you. Good luck!

DavidR is right. I've worked in radio and TV for years, and it takes mass or isolation to damp sound transmission. Padded walls will only dampen high-frequency reflected sound from inside the room.

But the best solution is to kill the noise at the source. There is nothing inherent in an electrical panel that should cause a buzz if it's working correctly.

One note: If you do add a second layer of drywall (and there are sound dampening drywall products available), run the seams between the panels at right angles to the existing ones. And pay close attention to sealing the joints around the edges.

I do not believe that sound would travel throught that padded wall I have used these in houses where the flex room was next to a bedroom and installed that padded wall to stop the T.V noise from bugging the person in the bedroom it was not perfect but sure help[ed and that was of cousre becuase the noise travelled out the doors too.And this is a nice stylish solution that anyone can do in a weekend

Foam padding might help some, but it's best used to deaden the room's own reverberation. To cut down noise transmission, you need a high-mass barrier, not a light, soft material.

See: Blockers vs absorbers.
Running the new drywall seams at right angles is a good idea. You can also offset the seams from the existing drywall's seams.

Sound dampening drywall has a sound barrier and isolation layer (I think similar to MLV) sandwiched between two gypsum board layers. It's a bit pricey, but here you're just covering one wall.

If you really like the looks of that padded wall, I guess you could hang a
layer of MLV behind it. But a second drywall layer instead of the padding will cut the noise better.
Also, be sure to check the fire spread characteristics of anything you put on the wall. Especially right behind your bed.

The foam provides the high mass barrier.

High mass barrier? How so? Foam is light, not dense, because it's mostly air.

Ask any good acoustical engineer what he or she recommends for blocking noise. It won't be foam padding.
Soundproofing myths

Sheets of lead. It works. Don’t let the fear mongers talk you out of it either.

Foam doesn’t do diddly for outside the room getting in. It is half decent for the usual bedroom noise escaping the bedroom. Even then, a solid door with little gap works better.

if the hum is from the electrical panel it might be high frequency vibration transmitted from the panel enclosure to the wall it's mounted to resulting in structure borne noise on your side. Putting isolators between the panel and the wall it's mounted on might reduce the structure-borne noise and would be pretty inexpensive.

Sophie is right, lead is the traditional solution, and is still considered the best. It's especially useful where you need a very thin and conformable noise barrier. Its downsides are toxicity and cost.

You can buy lead sheet on rolls. The 1/64" lead costs 3-5 times as much per square foot as 1/8" MLV, has the same mass (1 lb per square foot), and is rated the same for sound transmission class (STC 26).

DavidR has provided a lot of good info. You should be able to lessen the sound level coming through the wall by following some of the solutions provided above, but you may still end up with some residual, annoying sound. That is because: 1) you may not be able to cut out 100% of the problem, even with the best of solutions, and 2) this type of noise/hum/vibration may also be transferred to the floor, depending on the construction.

Lead would definitely work, if thick enough. Foam won’t do jack. The best way, though, without losing huge floor space, and perhaps more economical than a lead wall, is to remove the drywall, install resilient channels with sound clips, insulate the studs, and install 3 layers of type X drywall (NOT lightweight) with Green Glue between. Mass loaded vinyl can also be added and will get you a point or so more of a STC rating in the assembly.

That should work well. The clips and channels do help. I'm far from an expert, so I don't know if it would be enough difference to warrant losing a layer of drywall, or the labor of tearing it out. Maybe.

Thing is, green glue is great stuff, but expensive. You need 3 tubes per 4x8 drywall sheet, at about $17 each, for a total of over $50 plus your drywall. Sound damping drywall is $50-something per sheet, at least last time I looked, so it's a little cheaper, and less installation labor to boot.

A layer of 1/8" MLV instead of the green glue would give you an STC 26 barrier at around $35 per 4x8 sheet plus the drywall. I might be inclined to use one layer of green glue, for the isolation, and one layer of MLV. You can also get MLV with a closed cell foam back to add at least some of the isolation you'd otherwise get from green glue. It's more expensive, but not MLV-plus-green-glue more expensive.

The most economical fix of all is to stop the hum at its source, either by replacing the humming component with a quiet one, or by isolating it with resilient mounting. But that might require more cooperation from the condo association.

Green glue is way cheaper than the special drywall (which is actually the same thing but premade, thinner, and less effective).

Their website has a ton of technical documents, some of which are really good for consumers.

MLV isn’t that effective in a wall assembly. The marketers lie through deception. MLV is worth around 3 additional STC points in a wall assembly. Green glue is worth 15 stc or more alone.

I soundproofed slightly around the master bathroom. My husband keeps asking if I am using our jetted tub. (3-5x per week!) And he didn’t hear my screaming in the middle of the night when I broke my foot. Lol. Green glue is pretty darned effective.