The 10-year-old daughter of an American hostage has a fantasy.

Each day, when she boards a school bus to go home, she imagines her father will be there to greet her. Then she gets upset, realizing that her wish will once again go unfulfilled.

"She handles it by not coming in the house. She'll play outdoors for a while," says a relative. "It's a fantasy that he's not abandoned her."

For the families of the 53 hostages, the psychic strain of the past five months has been severe.

Ernest Cooke's mind starts to wander when he grades his students' papers. "It's harder to concentrate than ever," says Cooke, a marketing professor at Memphis State University in Tennessee. His son, vice consul Donald J. Cooke, 25, is a hostage.

"Quite frankly, I let go of some of my aggressions by kicking a wall," Cooke adds.

One hostage's wife told a friend that she thought she was having a heart attack -- an illusory ailment symptomatic of her emotional ordeal.

"I stay depressed all the time," said another captive's wife, who begged not to be identified for fear she would be hounded by the news media. She has rejected government offers of psychiatric counseling, doubting that it would help. She resents the periodic telephone calls from volunteer State Department workers.

"They ask, 'How are you doing?' -- which is a foolish question. The State Department stinks," she said. "At times, I wish they wouldn't ever call me at all."

Dorothy Limbert speaks about the families' feelings of loss and uncertainty from a unique vantage point. She is both a psychotherapist and the mother of a hostage, John W. Limbert Jr., 37, a State Department political officer.

"It's almost like a grieving situation with parts of it unresolved," she said in an interview at her Baltimore apartment. "You think about it constantly. You feel the hopelessness, the helplessness. It was difficult to concentrate at the beginning. It's difficult to finish a task."

Partly to help cope with her anxieties, Limbert has kept busy at work, as have other hostages' relatives. "There's the professional self and then there's the emotional self," she said. "It's like the actress' saying the show goes on. There's the professional peace."

Some of the hostages' families have proven remarkably resilient, buoyed by personal and community ties, religious faith and inner strength. Other captives' relatives are closer to despair.

But as the Iran crisis drags on, virtually all the families are finding it increasingly difficult to cope, according to clergymen and mental health specialists who have spoken with them.

"There is a real fraying of nerves," said the Rev. M. William Howard Jr., president of the National Council of Churches. He was among three clergymen who held Christmas services at the U.S. Embassy in Tehran and has kept in touch with a number of hostages' families since.

A mental health specialist who has talked with relatives of the hostages said he detected increasing disenchantment, impatience and anger with the Carter administration's handling of the Iran crisis. "The families are facing a lot of personal agony," he said. "It's the uncertainty . . . the tantalizing kind of thing."

The State Department, which has never faced a crisis of similar psychological dimensions, is considering an experimental program of sending psychiatric social workers to visit some hostages' families.

A report prepared for the State Department has spotlighted the issues. The hostages' families "are suffering as much if not more than the hostages themselves," warned the report by a task force headed by Charles R. Figley, director of Purdue University's Family Research Institute.

Partly because of their emotional stress, the hostages' families recently set up a group called FLAG (Family Liaison Action Group). "It's for psychological reasons as much as for anything else," said Louisa Kennedy, FLAG's representative to the news media and the wife of a hostage, economic officer Moorhead C. Kennedy Jr. "The release of the hostages is not going to bring an end to this for the families."

One psychiatrist who met with FLAG members is Steven Pieczenik, an authority on hostage psychology. In an interview, Pieczenik said that, in general, the hostages' families suffer from an underlying psychological conflict. They are caught between gratitude for U.S. efforts to gain the hostages' release and anger that these efforts have failed. As a result, he added, "one feels guilty if one doesn't feel grateful."

Although the hostages have frequently been compared with Vietnam war prisoners, psychologists point to significant differences. In contrast to the bitter controversy stirred by Vietnam, they note, the Iran crisis has prompted widespread American support for the captives and their families. For some families, the outpouring has been psychologically helpful.

Other hostages' families, however, have sought to seclude themselves from their communities and the media. Their reasons are usually sound, according to clergymen who have spoken with them. Some of these families want to protect their children from possible media attention, they say. Others are afraid of making an inadvertent public statement that might jeopardize a hostage's safety.

Yet for some hostages' families, talking provides an emotional outlet. "When I'm lecturing [in a university classroom], particularly, that's great therapy," Professor Cooke said. He even likes to talk to newspaper and television reporters, he added. "It helps to talk about it, for me at least."

The harshest recent blow for the hostages' families appears to have been the collapse of the U.N. commission's efforts to break the U.S.-Iran impasse. "This period that we're in right now is going to be the hardest that they've faced so far, because no one can give any indication of when it's going to end," said Auxiliary Bishop Thomas J. Gumbleton of Detroit, one of the clergymen who held Christmas services for the hostages.

"The real frustration is that there's nothing to do right now," remarked Dorothea Morefield, who husband, Consul General Richard H. Morefield, 50, is a hostage. "If someone had said in November that it would go on this long, it would be too much to handle."