The antelopes aren't playing. On this soundless afternoon, they are at rest. They are grazing in a bourn of wilderness in a part of south-central Wyoming where horizons are so vast that they bring on trances of peacefulness.

Here is one place, it is thought in the fullness of relief, where problem- free living appears to have a fair chance of thriving. Unsolvable problems are for the cities, not Wyoming, a state that is 49th in population and with little urban tumult.

Look again. Conflict and confusion are riding the range. The style is not that of a Butch Cassidy and his Powder River boys, whose freebootery helped give Wyoming its image of outlaw wildness a century ago. Today the shoot-outs are between citizens' groups and ranchers, environmentalists and politicians, ways of life against ways of progress.

One of the most tension-ridden parts of the state is Carbon County, an area of more than 8,000 square miles, which makes it larger than five states. Carbon County Coal Company, an employer of 300, works the county's only underground mine. It also has only one customer -- the Northern Indiana Public Service Co. In 1978, the Indiana utility contracted with the coal company to supply it with 1.5 million tons of low-sulphur coal annually. Now, with cheaper coal available on the spot market and a two-year stockpile of $46-a-ton Carbon County coal, the utility and the company are in court in a contract dispute. The utility says stop the shipments. The company says honor the contract.

For three weeks courts in Wyoming and Indiana have been involved. In Cheyenne, the voice of rangeland integrity was heard statewide when a district court judge accused the Indianans of thinking that they had "a dumb Wyoming judge" who could be duped into agreeing that the contract was no longer binding because conditions had changed.

Much more than the language of a contract is involved. The fate of one town -- Hanna -- is tied to the company with one customer. If the mines fail, so does the town, the mayor has said. Hanna would not have the $1.8 million in taxes the company paid in 1984. The town's 27 employees would be laid off. Some 220 children would have no school.

Wyoming's veins run deep with a fourth of the nation's coal reserves. But it has far less of America's concern when places such as Hanna are threatened with ghost-town futures. Contractual problems are common throughout Wyoming. Coal mine shutdowns here are becoming as routine as bank closings in Iowa.

The aura of prairie timelessness has little to do with the speed of economic change. Only seven years ago, a Washington Post reporter went to Hanna and other towns. He wrote, accurately at the time, that Carbon County, because of the nation's demand for low- sulphur coal, was "one big boom town." Now it is becoming a bust town. The labor force in January 1984 was 9,649. A year later it dropped to 8,968.

While the remaining citizens brace for the next economic assault, wildlife is also under siege. Ranchers and rancher-dominated legislatures, as is the case in Wyoming, now see the West facing ruination not from wild coal prices but wild horses. An estimated 52,000 free-roaming mustangs and other breeds are ranging on the public lands of Wyoming and other states. The argument for killing them -- many would end up as dog food -- is that wild horses cause forage damage. In fact, it is more the bumbling policies of the Bureau of Land Management, as well as the reality that many ranchers lack land-use skills, that are depleting the pasturelands.

Roped by the ranchers -- and around the neck at that -- Congress appropriated $17 million last year to round up 17,000 horses and burros. The Humane Society of America reports that the funds "provide three times more money than Congress has ever set aside in a single year for this purpose." It blows away the protections of the 1971 Free-Roaming Wild Horse and Burro Act.

As parts of Wyoming are losing people, jobs, mines and horses, the response of Congress and the Reagan administration is to bring on the MX missiles. When the eastern media report on Washington's debates on the missiles, it is rarely mentioned that many of these bombs of annihilation are to be stored in the earth of Wyoming. One despairing Carbon County ranchwoman said of the Minuteman missiles now in the ground and the MXs planned as replacements, "They are so close that we put them out of our mind."

Western Solidarity, a group working to defeat the MX, has roused much of Wyoming to opposition. The trouble is that threats to the economy and the land press in also. A new state motto might be needed: Where do we begin? "The aura of prairie timelessness has little to do with the speed of economic change."