Washington — Washington is beginning to expect miracles of George Shultz. In his first months as secretary of state, Mr. Shultz got a grip on Middle East problems and helped to shape a policy that restored negotiating momentum.
He then moved on to defuse tensions with Western Europe over the controversial Soviet-European gas pipeline. Next, he managed to prevent, at least temporarily, what could have developed into a trade war with the Europeans.
Shultz is now expected by many experts in the bureaucracy to dominate two huge new problem areas: arms-control negotiations with the Soviet Union and international economic policy. An economist, he is already deeply engaged on the economic front. He has been getting briefed on the arms-control issues for some months now.
Some officials are beginning to worry, however, that too much is expected of the hard-working secretary of state. In an administration which has not been heavily endowed with talent in the foreign policy field, Shultz stands out despite his low-key style.
As a member of the Senate Foreign Relations Committee staff put it, Shultz has become the administration's ''Mr. Fix-it.''
''What this administration needs is someone who can pull all the strands of international economic policy together,'' said the same Senate specialist. ''Whether he likes it or not, Shultz is being drawn into this. He's the one who's expected to pull order out of chaos.''
The same thing is being said by some officials in the arms-control field.
Even as he moves into these new fields, Shultz cannot simply leave other major issues such as the Middle East behind. Aides say he places the Middle East at the top of his list of priorities, alongside US-Soviet relations, arms-control negotiations, and European security issues. But they say that Shultz has moved international economic problems to the top as well. He is planning a trip to East Asia shortly, and Central America continues to demand a degree of attention.
Shultz clearly has a full year ahead, and some observers are asking: How can one man do it all?
The answer is that Shultz does not try to do it all. He is secure enough to delegate considerable responsibility. The day-to-day handling of the simmering Lebanon crisis, for example, is done by Shultz's deputy, Kenneth Dam, who heads an interagency committee on Lebanon. The secretary also respects the professional diplomats who are working for him in field and relies heavily on them.
Shultz has often remarked that the Middle East could easily chew up all of a secretary of state's time. Although a trip to the region could come later this year, Shultz has so far avoided the temptation to travel to the Middle East. So far, he has traveled a good deal less than any of his recent predecessors. He no doubt noted the toll which fruitless shuttles to Argentina took on his immediate predecessor, Alexander M. Haig Jr. One of the most widely traveled secretaries of state, Cyrus R. Vance, ended his term virtually exhausted.
Shultz conveys the impression that he is moving purposefully and methodically , but with plenty of time for subordinates. Officials in the State Department's Bureau of Human Rights and Humanitarian Affairs, which did not win many battles with Mr. Haig, or merit much of his attention for that matter, were delighted when Shultz dropped in unannounced recently to say a word or two to each and every staff member who happened to be on duty at the time. The bureau had been waiting for months for much-needed renovations. The day after Shultz's visit, workmen arrived to carry out those renovations.
More recently, Shultz was seen having lunch in the State Department cafeteria. Lower-ranking employees, not used to rubbing shoulders with a secretary of state, saw this as one more small sign that Shultz identifies with them. And it was not the first time that Shultz had visited the cafeteria.
Although the seemingly unhurried secretary of state does not act like a man who carries a tremendous burden of responsibility, that burden is heavy indeed. One need only take a look at the two new top priority problem areas which he faces as he begins this new year:
* US-Soviet relations inevitably find their way to the very top of the list of top priorities of any secretary of state. Shultz is being criticized by some officials for not moving more rapidly to give new direction to US arms-control negotiations. There is a feeling that the US is losing the propaganda battle over the planned deployment of new American nuclear missiles in Western Europe. A flurry of statements and seemingly new arms-control proposals emanating from the new Soviet leader, Yuri Andropov, have grabbed recent headlines.
But beyond the realm of propaganda,some officials are said to believe that unless the US shows a greater willingness to be flexible about President Reagan's ''zero-option'' proposal for cuts in medium-range nuclear missiles in Europe, the NATO alliance consensus on the need to deploy new missiles could collapse. Paul Nitze, the chief US negotiator to the Geneva talks on this subject, is said to favor greater flexibility.
Arms-control is not a field in which Shultz has had great experience, and he is taking his time asserting himself. But officials point out that Eugene V. Rostow, the head of the Arms Control and Disarmament Agency has been weakened both by illness and by opposition in Congress to some of his staff appointments. National security adviser William P. Clark, a novice in the field, does not seem to be playing a strong coordinating role.
The Defense and State Departments are divided on the issue of flexibility, and Defense Secretary Caspar Weinberger's comments on the subject of nuclear warfare have tended to frighten Europeans. It is a situation which cries out for attention from Mr. Shultz.
* In the international economics field, Shultz is now highly active. He is possibly the most experienced top-level official in the administration in this field, and he is sitting in on meetings of key policymakers who are dealing with domestic economic and budgetary problems as well.
Under Shultz, the State Department is taking the lead in planning for a new allied economic summit meeting to be held at the end of May in Williamsburg, Va. Shultz and his advisers are determined to avoid anything resembling the disastrous misunderstanding over East-West trade which followed the Versailles summit in June of last year.
The State Department experts want to shift the style of these annual summits away from the Versailles-type extravaganza, which raises unrealistically high expectations, and back to the original concept, which was to hold less rigidly structured, lower-key consultations among the allied leaders. In the Shultz view , the latter type of summit may not produce headline-grabbing results, but it can greatly increase understanding on all sides.
Officials say that in every economic crisis area - whether it be a question of exchange rates or the developing countries' debts - Shultz wants to keep the main focus on one overriding issue: How to bring about worldwide economic recovery.''We've still got the Middle East and all the other high-priority items ,'' said one State Department official.
''We've got people killing each other in some parts of the world. But there's a lot of underlying tension and frustration in the economic area that could explode in the wrong way as well.''